\r\n\tThe outcome of cancer therapy with radiation has been improving over the years due to technological progress. However, due to the biological property of cancer, current radiotherapy has limitations. Therefore, in consideration of the dynamics of tumor cells caused by radiation irradiation, attempts are being made to overcome the current drawbacks and to improve radiotherapy. It is expected that carbon ion beams, hyperthermia, oxygen effect, blood flow control, etc. will be used in the future in order to improve the treatments. This book aims to introduce research results of various radioprotective agent development research and hypoxia sensitizers.
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\n
1. Introduction
\n
The proposed chapter focuses on the methods and applications of the graph theory in the area of quantum transport on combinatorial and metric (often referenced to as “quantum”) graphs. It is well known that perfectly periodic potentials in Euclidean spaces or on periodic lattices create favorable conditions for nonlocalized solutions of Schrödinger and/or wave equations. In quantum physics, this results in transport of quantum particles, for example, electrons or phonons. However, after the seminal, Nobel prize winning work [1] published by Philip W. Anderson in 1958, it has been realized by physicists that the propagation of quantum particles in an imperfect environment, modeled by a random or almost periodic electrostatic or magnetic potential, can be significantly inhibited, to the point where mobile quantum particles, e.g., electrons, are localized: their wave functions decay exponentially away from some loci— their respective localization centers. In many applications, the media where quantum particles propagate are not periodic crystals, but have instead a structure of more or less complex graphs formed by atoms, and therefore are treated as disordered media. The structural disorder can be complemented by a parametric one, e.g., in the context of weighted graphs where the canonical graph Laplacian \n\n\n\nΔ\nΓ\n\n\n\n on \n\nΓ\n\n is modulated by variable weights assigned to the edges. In general terms of the spectral theory of unordered structures, this is an instance of the “off‐diagonal” disorder. Furthermore, it is important to analyze the spectral properties of the discrete analogs of the Schrödinger operators \n\n\n−\n\nΔ\nΓ\n\n+\nV\n\n\n on a graph \n\nΓ\n\n, where \n\nV\n\n is a fixed or random real‐valued function on \n\nΓ\n\n.
\n
The recent wake of interest to the nanosystems and molecular devices attracted many researchers to such models, and numerous intriguing problems in this area still remain challenging and wide open. It has been understood that the classical aspects of the graph theory, such as isoperimetric estimates (particularly, the Cheeger bounds) and deep results of the spectral theory of graphs, are of great importance to the localization/delocalization processes on graphs other than periodic lattices embedded in a Euclidean space.
\n
Furthermore, the most recent developments in the spectral theory of disordered quantum systems, initiated independently and simultaneously by physicists (cf., e.g., [2]) and mathematicians (cf. [3, 4, 5, 6], emphasized the role of interparticle interaction which had been consciously and explicitly neglected in the pioneering works due to the complexity of the analysis involved, although P. W. Anderson himself was concerned about possible effects of interaction on the fundamental properties of the quantum transport. Following the first mathematical works in this direction, the notion of a multiparticle quantum graph has been recently introduced by Sabri [7].
\n
Summarizing, the proposed chapter provides to the reader an overview of synthetic techniques and results where the traditional problem of the combinatorial and spectral graph theory is intertwined with complementary structures, ideas, and methods of functional analysis and quantum mechanics, in response to the new challenges in modern technology.
\n
\n
\n
2. Isoperimetric bounds, spectral gaps, and quantum localization
\n
The integer lattices \n\n\n\nZ\nd\n\n\n\n, \n\n\nd\n≥\n1\n\n\n, endowed with the usual graph structure constitute a very particular class of connected graphs. The spectra and (generalized) eigenfunctions of their canonical graph Laplacians are easy to find, due to the commutative group structure of these lattices. The lattice Laplacian \n\n\n\nΔ\n\n\nZ\nd\n\n\n\n\n\n is the canonical Laplacian on \n\n\n\nZ\nd\n\n\n\n endowed with the graph structure where the edges are formed by the pairs \n\n\n\n(\n\nx\n,\ny\n\n)\n\n\n\n with Euclidean distance \n\n\n\n\n\n|\n\nx\n−\ny\n\n|\n\n\n2\n\n=\n1\n\n\n. It follows from the Fourier analysis on the additive group \n\n\n\nZ\nd\n\n\n\n that the spectral measure of \n\n\n\nΔ\n\n\nZ\nd\n\n\n\n\n\n is absolutely continuous (with respect to the Lebesgue measure). The graph Laplacians \n\n\n\nΔ\nΓ\n\n\n\n are defined as nonpositive operators, but in mathematical physics one considers \n\n\n−\n\nΔ\nΓ\n\n\n\n instead. The spectrum of \n\n\n−\n\nΔ\n\n\nZ\nd\n\n\n\n\n\n is easily computed, knowing that its Fourier image is the operator of multiplication by the function
The generalized eigenfunctions are given by the respective Fourier harmonics (plane waves) \n\n\n\nx\n\n↦\nexp\n\n(\n\ni\n\n(\n\n\np\n\n,\n\nx\n\n\n)\n\n\n)\n\n\n\n, where \n\n\n\n(\n\n\np\n\n,\n\nx\n\n\n)\n\n=\n\np\n1\n\n\nx\n1\n\n+\n…\n+\n\np\nd\n\n\nx\nd\n\n\n\n. In physics, one often works with finite cubes, where the eigenfunctions of the respective Laplacian are combinations of plane waves. Particular questions concerning the Laplacians relative to such finite graphs depend upon specific intended applications. A number of situations give rise to the quantitative analysis of a few of the lowest eigenvalues of \n\n\n(\n−\n\nΔ\nΓ\n\n)\n\n\n, numbered in increasing order. In particular, the spectral gap and the analytic form of the eigenfunctions with eigenvalues \n\n\n\nλ\n0\n\n\n\n and \n\n\n\nλ\n1\n\n\n\n, known explicitly for the rectangles, are more difficult to analyze for general graphs. One possible question is about the size of the gap \n\n\n\nλ\n1\n\n−\n\nλ\n0\n\n\n\n: it features a power‐law decay as the size \n\nL\n\n of the cube \n\n\n\n\n\n[\n\n1\n,\nL\n\n]\n\n\nd\n\n\n\n grows, but what can one say about the spectral gap for less regular graphs? Furthermore, it is readily seen that the eigenvalue \n\n\n\nλ\n1\n\n\n\n is degenerate on the cube, and has multiplicity equal to the dimension \n\nd\n\n: the respective eigenfunctions are the lowest‐frequency harmonics, related to the global geometry of the cube, but the situation for general graphs is more complex.
\n
Before going to the answers, provided by the graph theory for a large class of nonperiodic graphs, we give some motivations coming from the spectral theory of random operators.
\n
One remarkable phenomenon relative to disordered media was discovered in the 1960s by physicist I.M. Lifshitz [8] and colorfully called in the physics and mathematics communities “Lifshitz tails”: the eigenvalue distribution of the random operators \n\n\n\nH\nV\n\n=\n−\nΔ\n+\nV\n\n(\nω\n)\n\n\n\n decays extremely as the energy \n\nE\n\n approaches the lower edge of spectrum.
\n
In this chapter, we always assume the random potential to take i.i.d. (independent and identically distributed) values. In addition, we assume the common probability distribution of these random values to admit a bounded probability density.
\n
We shall use the following notions and notations. Given a potential \n\n\nV\n:\n\nZ\nd\n\n→\nR\n\n\n, we consider the discrete Schrödinger operator \n\n\n\nH\nV\n\n=\n−\nΔ\n+\nV\n\n\n, where \n\nΔ\n\n is the graph Laplacian on the integer lattice \n\n\n\nZ\nd\n\n\n\n. Further, for each integer \n\n\nL\n≥\n1\n\n\n and a lattice point \n\nx\n\n denoted by \n\n\nB\n\n(\n\nL\n,\nx\n\n)\n\n\n\n the cube centered at \n\nx\n\n of side length \n\n\n2\nL\n\n\n, and let \n\n\n\nH\nL\n\n\n\n be the Schrödinger operator \n\n\n−\n\nΔ\n\nB\n\n(\n\nL\n,\n0\n\n)\n\n\n\n+\nV\n\n\n in the cube \n\n\nB\n\n(\n\nL\n,\n0\n\n)\n\n\n\n; here, \n\n\n−\n\nΔ\n\nB\n\n(\n\nL\n,\n0\n\n)\n\n\n\n\n\n is the canonical graph Laplacian in \n\n\nB\n\n(\n\nL\n,\n0\n\n)\n\n\n\n with the graph structure inherited from the integer lattice, where the edges are given by the pairs of nearest neighbours in the Euclidean distance. Next, denote by \n\n\n\nλ\nk\n\n\n\n the eigenvalues of \n\n\n\nH\nL\n\n\n\n numbered in increasing order, and introduce the finite‐volume eigenvalue counting function
whenever it exists. Otherwise, we say that the limiting distribution function does not exist.
\n
In the case where a fixed potential \n\n\nV\n:\n\nZ\nd\n\n→\nR\n\n\n is replaced by a sample \n\n\nV\n\n(\nω\n)\n\n\n\n of a random field on the lattice, the operator \n\n\n\nH\nV\n\n=\n\nH\nV\n\n\n(\nω\n)\n\n\n\n also becomes random.
\n
In physical terminology, widely used also in mathematical physics of disordered media, \n\n\nN\n\n(\nE\n)\n\n\n\n is usually called the integrated density of states (IDS). The existence of the above limit is not obvious and, generally speaking, the limit may not exist. However, the existence of \n\n\nN\n\n(\nE\n)\n\n\n\n for any energy can be established by the methods of ergodic theory in a particular, but very rich and useful for physical applications class of ergodic operators (including all i.i.d. potentials), as well as for the periodic and almost‐periodic potentials. In fact, the latter classes can be incorporated into the general scheme of ergodic operators; cf., e.g., the monographs [9, 10]. Moreover, the IDS for ergodic potentials is nonrandom; in physical terminology, IDS is a “self‐averaging” quantity. Simply put, spatial average coincides with the ensemble average for ergodic operators.
\n
Whenever the potential \n\nV\n\n of the Schrödinger operator \n\n\n\nH\nV\n\n=\n−\nΔ\n+\nV\n\n\n is lower bounded, e.g., nonnegative, \n\n\n\nH\nV\n\n,\n\n\n and its spectrum \n\n\nSpec\n\n(\n\n\nH\nV\n\n\n)\n\n\n\n have the same property, since\n\n\n−\nΔ\n\n\n is nonnegative. Therefore, \n\n\n\nE\n0\n\n:\n=\ninf\n\nSpec\n(\n\nH\nV\n\n)\n>\n−\n∞\n\n\n. In physics, \n\n\n\nE\n0\n\n\n\n is called the ground state energy. A number of important quantities and phenomena are related to the ground state energy and also to the behavior of the IDS as the energy \n\nE\n\n approaches \n\n\n\nE\n0\n\n\n\n. Lifshitz [8] discovered that for a large class of Hamiltonians with random potential energy, including random Schrödinger operators \n\n\n\nH\nV\n\n=\n−\nΔ\n+\nV\n\n(\nω\n)\n\n\n\n on a lattice, the IDS decays very fast as \n\n\nE\n↓\n\nE\n0\n\n\n\n: for some \n\n\n\nC\n1\n\n,\nC\n>\n0\n\n\n
Lifshitz tails have numerous important ramifications in theoretical and experimental physics. They also result in a nonperturbative onset of the Anderson localization on lattices for any, arbitrarily small amplitude of the random potential \n\n\nV\n\n(\nω\n)\n\n\n\n. Away from the spectral edge, the proofs of localization require a sufficiently large amplitude of \n\n\nV\n\n(\nω\n)\n\n\n\n; moreover, it is widely believed that in dimension \n\n\nd\n≥\n3\n\n\n, in the models where the random potential \n\n\ng\nV\n\n(\nω\n)\n\n\n\n has a sufficiently small amplitude \n\n\n\n|\ng\n|\n\n\n\n, there are intervals \n\nI\n\n of energy where the corresponding generalized eigenfunctions (“extended quantum states”) are not square‐summable, and the spectral measure has in \n\nI\n\n a nontrivial absolutely continuous component. In simpler terms, there is a nontrivial quantum transport in some energy zones.
\n
A substantial progress has been achieved in the direction of proofs of localization near the spectral edge (or edges). For a long time, most of the efforts have been made in the analysis of lattice Hamiltonians \n\n\n\nH\nV\n\n=\n−\nΔ\n+\nV\n\n(\nω\n)\n\n\n\n. Recently, it has been shown in Ref. [11] that a number of results obtained on the integer lattices can be extended to much more general graphs of polynomial (or, more generally, subexponential) growth. The key point is the availability of lower bounds on the spectral gap in terms of the Cheeger’s constant of the graph.
\n
Consider a lattice cube \n\n\nB\n=\nB\n\n(\n\nL\n,\n0\n\n)\n\n\n\n with the graph structure inherited from the lattice, and the random lattice Schrödinger operator \n\n\n\nH\n\nB\n,\nV\n\n\n\n(\nω\n)\n\n=\n−\n\nΔ\nB\n\n+\nV\n\n(\nω\n)\n\n\n\n on it. The starting point of the localization analysis of this finite‐volume Hamiltonian is the estimate of the probability to have some eigenvalues of \n\n\n\nH\n\nB\n,\nV\n\n\n\n(\nω\n)\n\n\n\n in a small interval \n\n\n\nI\nϵ\n\n=\n\n[\n\n\nE\n\n0\n,\n\n\n\nE\n0\n\n+\nϵ\n\n]\n\n\n\n near the spectral edge \n\n\n\nE\n0\n\n\n\n. This is closely related to the Lifshitz tails. One needs a finite‐volume analog of the limiting Lifshitz asymptotics, but for the localization analysis, one can settle for a sufficiently strong, albeit not necessarily sharp, upper bound on the probability
Proposition 1.Let\n\nA\n\nbe a self‐adjoint operator in a finite‐dimensional Hilbert space\n\nH\n\n, and\n\n\n\nE\n0\n\n=\ninf Spec\n\n(\nA\n)\n\n\n\nbe a simple eigenvalue. If a vector\n\nψ\n\nwith unit norm satisfies\n\n\n\n〈\n\nψ\n,\nA\nψ\n\n〉\n\n<\n\nE\n1\n\n:\n=\ni\nn\nf\nS\np\ne\nc\n\n(\nA\n)\n\n\\\n\n{\n\n\nE\n0\n\n\n}\n\n,\n\n\nthen
Now one can see the importance of the size of the lowest spectral gap, \n\n\nη\n=\n\nE\n1\n\n−\n\nE\n0\n\n\n\n. As was mentioned above, \n\nη\n\n is easily calculated explicitly for the Laplacians in lattice rectangles, but of course there is no universal formula for general finite graphs. The following result was obtained in Ref. [11].
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Proposition 2.Let be given a finite connected subgraph\n\nG\n\nof a locally finite countable connected graph\n\nΓ\n\nsatisfying the following condition: there exists some real constants\n\n\nd\n≥\n1\n\n\nand\n\n\nC\n>\n0\n\n\nsuch that any ball\n\n\nB\n\n(\n\nL\n,\nx\n\n)\n\n⊂\nΓ\n\n\nof radius\n\n\nL\n≥\n1\n\n\nhas cardinality
Apart from the canonical negative Laplacian \n\n\n(\n−\n\nΔ\nG\n\n)\n\n\n, it is often more convenient to work with its modified variant \n\n\n\nL\nG\n\n\n\n defined by
where \n\n\n(\n\nn\nG\n\n(\nx\n)\n\n\n and \n\n\n(\n\nn\nG\n\n(\ny\n)\n\n\n are the coordination numbers of the vertices \n\nx\n\n and \n\ny\n\n, respectively: \n\n\n\nn\nG\n\n\n(\nx\n)\n\n=\ncard\n\n(\n\nB\n\n(\n\n1\n,\nx\n\n)\n\n\\\n\n{\nx\n}\n\n\n)\n\n\n\n. The coordination numbers are nonzero, whenever the graph is connected and has more than one vertex. Below we quote the bounds obtained for the modified Laplacian, but, up to some constants, they remain valid for the original Laplacian.
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Definition 2.The Cheeger’s constant of a finite connected graph\n\nG\n\nis the following quantity:
where the minimum is taken over all nontrivial partitions\n\n\nG\n=\nW\n⊔\n\nW\nc\n\n\n\nof the graph\n\nG\n\ninto disjoint subgraphs\n\nW\n\nand its complement\n\n\n\nW\nc\n\n\n\n.
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Denote by \n\n\n\nμ\nk\n\n\n(\nG\n)\n\n\n\n, \n\n\nk\n≥\n0\n\n\n, the eigenvalues of \n\n\n\nL\nG\n\n\n\n numbered, as their counterparts \n\n\n\nλ\nk\n\n\n(\nG\n)\n\n\n\n, in increasing order. Then one has the following results (cf. [13]).
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Proposition 3. Let be given a finite connected graph \n\nG\n\n of diameter \n\n\n\nD\nG\n\n:\n=\ndiam\nG\n\n\n. Let \n\n\n\nμ\n1\n\n\n(\nG\n)\n\n\n\n be the first nonz ero eigenvalue of the modified graph Laplacian \n\n\n\nL\nG\n\n\n\n, and \n\n\nh\n\n(\nG\n)\n\n\n\n the Cheeger’s constant of \n\nG\n\n. Then
For a finite connected subgraph \n\n\nG\n⊂\nΓ\n\n\n satisfying the growth condition in Eq. (7), one has \n\n\n\nD\nG\n\n≤\n\n|\nG\n|\n\n−\n1\n\n\n and \n\n\nvol\n\n(\nG\n)\n\n≤\n\nC\nd\n\n|\nG\n|\n\n\n. Combined with the inequalities in Eqs. (7) and (9), this results in the following lower bound for \n\n\n\nE\n1\n\n\n(\nG\n)\n\n\n\n:
The upper bound by \n\n\n2\nh\n\n(\nG\n)\n\n\n\n is not quite explicit in general (this depends of course on the specific problems at hand), but for finite connected subgraphs \n\n\nG\n⊂\nΓ\n\n\n one can prove that \n\n\n\n\nlim\n\n\n\n|\nG\n|\n\n→\n+\n∞\n\n\n\nμ\n1\n\n\n(\nG\n)\n\n=\n0\n\n\n (cf. [14]). Now we are ready to prove the following result.
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Lemma 1. Let \n\nG\n\n be a finite connected subgraph of a graph \n\nΓ\n\n satisfying the growth condition in Eq. (6), and \n\n\n0\n<\nη\n≤\n\n1\n6\n\n\nλ\n1\n\n\n(\n\n−\n\nΔ\nG\n\n\n)\n\n\n\n. Let \n\nV\n\n be a nonnegative real function on \n\nG\n\n, and set \n\n\n\nV\nη\n\n(\nx\n)\n:\n=\nmin\n[\nV\n(\nx\n)\n,\n2\nη\n]\n,\n\nH\nG\n\n=\n−\n\nΔ\nG\n\n+\nV\n.\n\n\n Then \n\n\n\nE\n0\n\n\n(\n\n\nH\nG\n\n\n)\n\n≥\n\n1\n2\n\n\n\n\n|\nG\n|\n\n\n\n−\n1\n\n\n\n\n∑\n\nx\n∈\nG\n\n\n\n\nV\nη\n\n\n\n\n(\nx\n)\n\n.\n\n\n
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Proof. Consider the normalized eigenfunctions of \n\n\n\nΔ\nG\n\n\n\n with the eigenvalue \n\n\n\nE\n0\n\n\n\n, viz. \n\n\n\nψ\n0\n\n=\n|\nG\n\n|\n\n−\n1\n/\n2\n\n\n\n\n1\n\nG\n\n\n\n. Next, introduce an auxiliary operator \n\n\nK\n=\n−\n\nΔ\nG\n\n+\n\nV\nη\n\n\n\n. By nonnegativity of the functions \n\n\n\nV\nη\n\n≤\nV\n\n\n, we have the inequalities (in the sense of the associated quadratic forms)
so that by the min‐max principle, we have \n\n\n\nE\n0\n\n\n(\n\n\nH\nG\n\n\n)\n\n≥\n\nE\n0\n\n\n(\nK\n)\n\n\n\n and \n\n\n\nE\n1\n\n\n(\n\n\nH\nG\n\n\n)\n\n≥\n\nE\n1\n\n\n(\nK\n)\n\n≥\n\nE\n1\n\n\n(\n\n−\n\nΔ\nG\n\n\n)\n\n\n\n. Since \n\n\n\nE\n0\n\n=\n0\n\n\n, it follows that \n\n\n\nΔ\nG\n\n\nψ\n0\n\n=\n0\n\n\n, and therefore,
Thus, we have Temple’s inequality to \n\n\n\nψ\n0\n\n,\nK\n\n\n and \n\n\n\nE\n1\n\n\n(\nK\n)\n\n\n\n. Note first that by \n\n\n\nΔ\nG\n\n\nψ\n0\n\n=\n0\n\n\n, one has
Lemma 2. Consider a nonnegative i.i.d. random field \n\n\nV\n\n(\n\nx\n,\nω\n\n)\n\n\n\n on a locally finite graph \n\nΓ\n\n satisfying the growth condition in Eq. (7), with the common marginal probability distribution function \n\n\nF\n(\nt\n)\n:\n=\n\nP\n\n{\nV\n(\nx\n,\nω\n)\n≤\nt\n}\n\n\n, and assume the following:
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There exist arbitrarily large \n\n\nL\n∈\nN\n\n\n such that any ball \n\n\nB\n\n(\n\nL\n,\nx\n\n)\n\n⊂\nΓ\n\n\n can be partitioned into connected graphs \n\n\n\nG\ni\n\n\n\n with \n\n\n\nL\n\n\nδ\n\n12\n\n\n\n\n≤\n\n|\n\n\nG\ni\n\n\n|\n\n≤\n\nL\n\n1\n/\n12\n\n\n\n\n, for some \n\n\nδ\n∈\n\n(\n\n0\n,\n1\n\n)\n\n\n\n.
\n\n\n\nF\n\n(\n\nt\n+\ns\n\n)\n\n−\nF\n\n(\nt\n)\n\n≤\nC\n\ns\nβ\n\n\n\n for some \n\n\nβ\n∈\n\n\n, \n\n\nC\n>\n0\n\n\n and all \n\n\nt\n∈\nR\n,\ns\n>\n0\n\n\n;
Then for any positive integer \n\nn\n\n, there exists a finite connected subgraph \n\n\nG\n⊂\nΓ\n\n\n with \n\n\nG\n∨\n≥\nn\n\n\n and satisfying the following spectral bound:
Let \n\n\n\nη\nG\n\n=\n\n\n\nc\nd\n\n\n6\n\n\n\n\n|\nG\n|\n\n\n\n−\n2\n\n\n\n\n, then \n\n\n2\n\nη\nG\n\n≤\n\n1\n3\n\n\nE\n1\n\n\n(\n\n−\n\nΔ\nG\n\n\n)\n\n\n\n, hence we can use \n\n\nη\n=\n\nη\nG\n\n\n\n and get
The value \n\n\n\nη\nG\n\n\n\n can be made arbitrarily small by taking the cardinality of the graph \n\nG\n\n large enough, and by assumption on continuity of the probability distribution function \n\n\n\nF\nV\n\n\n\n, for \n\n\n\nη\nG\n\n\n\n sufficiently small we have \n\n\n\nF\nV\n\n\n(\n\n\nη\nG\n\n\n)\n\n≤\n1\n/\n4\n\n\n. Recall that the values of the random potential \n\n\nV\n\n(\n\nx\n,\nω\n\n)\n\n\n\n are i.i.d., and so are \n\n\n\nV\nη\n\n\n(\n\nx\n,\nω\n\n)\n\n,\n\n\n since they are functions of i.i.d. r.v., so the probability for the sample mean of a large number of i.i.d. random variables to take a value away from the expectation can be assessed with the help of the large deviations theory. Specifically, for any \n\n\nn\n≥\n1\n\n\n and i.i.d. r.v. \n\n\n\nξ\n1\n\n,\n…\n,\n\nξ\nn\n\n\n\n, for any \n\n\nη\n>\n0\n\n\n such that \n\n\n\nP\n\n\n{\n\n\nξ\n1\n\n≤\n2\nη\n\n}\n\n≤\n1\n/\n4\n\n\n, one has
(see the details in Ref. [11]). Furthermore, we have \n\n\n\n\n\n|\nG\n|\n\n\n\n−\n3\n\n\n≤\n\n\n\nc\nd\n\n\n6\n\n\n\n\n|\nG\n|\n\n\n\n−\n2\n\n\n\n\n for \n\n\n\n|\nG\n|\n\n\n\n large enough. Now the lower bound in Eq. (13) combined with Eq. (18) proves the claim:
Proof. Fix a vertex \n\n\nx\n∈\nΓ\n\n\n. By assumption (i), there are arbitrarily large \n\nL\n\n such that the ball \n\n\nB\n\n(\n\nL\n,\nx\n\n)\n\n\n\n can be partitioned into connected graphs \n\n\n\nG\ni\n\n\n\n with \n\n\n\nL\n\n\nδ\n\n12\n\n\n\n\n≤\n\n|\n\n\nG\ni\n\n\n|\n\n≤\nM\n\n(\n\nL\n,\nx\n\n)\n\n≤\n\nL\n\n1\n/\n12\n\n\n\n\n. The operator \n\n\n\n(\n\n−\n\nΔ\nG\n\n\n)\n\n\n\n admits the following lower bound in the sense of quadratic forms:
Observe that by (i), \n\n\n\nL\n\n−\n1\n/\n4\n\n\n≤\n\n\n\n|\n\n\nG\ni\n\n\n|\n\n\n\n−\n3\n\n\n≤\n\nL\n\n−\nδ\n/\n4\n\n\n\n\n. Owing to Lemma 1, we conclude that
provided that \n\nL\n\n is sufficiently large. \n\n□\n\n
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Now we give an application of the above result, which was the main motivation in Ref. [11], and explain how the bounds for the eigenvalues of the graph Laplacians give rise to the decay of eigenfunctions. Due to the size limitations of the present chapter, we treat the Green’s functions in finite cubes, but the decay of the latter is important in itself, for physical applications, and it is known to imply the decay of eigenfunctions (cf. [10]). The decay of the Green’s functions is established by the so‐called multiscale analysis (MSA), an inductive scaling algorithm which we will sketch now (details can be found in [9, 10]). First, fix some notations and give some definitions. Given a ball \n\n\nB\n=\nB\n\n(\n\nL\n,\nu\n\n)\n\n⊂\nΓ\n\n\n and the operator \n\n\n\nH\nB\n\n=\n−\n\nΔ\nB\n\n+\nV\n\n\n, we denote by \n\n\n\nG\nB\n\n\n(\nE\n)\n\n\n\n its resolvent operator \n\n\n\nH\nB\n\n\n\n, and by \n\n\n\nG\nB\n\n\n(\n\nx\n,\ny\n,\nE\n\n)\n\n\n\n, \n\n\nx\n,\ny\n∈\nB\n\n\n, the matrix elements of the resolvent (usually called Green functions) in the standard orthonormal delta‐basis formed by the single‐site indicator functions \n\n\n\n\n1\n\nx\n\n\n\n. Further, given a ball \n\n\nB\n⊂\nΛ\n\n\n inside a larger connected subgraph \n\n\nΛ\n⊂\nΓ\n\n\n, the Green functions satisfy an inequality, often called Simon‐Lieb inequality (sli), easily following from the second resolvent identity: for any \n\n\nx\n∈\nB\n\n\n and \n\n\ny\n∈\nΛ\n\\\nB\n\n\n, one has
Since \n\n\n\n|\n\nB\n\n(\n\nL\n,\nu\n\n)\n\n\n|\n\n≤\n\nC\nd\n\n\nL\nd\n\n\n\n, and the coordination numbers are uniformly bounded by \n\n\n\nC\nd\n\n\n1\nd\n\n=\n\nC\nd\n\n\n\n, we have \n\n\n\n|\n\n∂\nB\n\n|\n\n≤\n\nC\nd\n2\n\n\nL\nd\n\n\n\n, so the above GRI implies
Here, \n\nx\n\n is an arbitrary point of \n\n\nB\n=\nB\n\n(\n\nL\n,\nu\n\n)\n\n\n\n, but we will be mostly interested in the case where \n\n\nx\n=\nu\n\n\n, so the first maximum in the above RHS becomes a characteristic of the ball \n\nB\n\n.
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A simple but important observation is that when \n\n\nq\n:\n=\n\nC\nd\n2\n\n\nL\nd\n\n\n\nmax\n\n\nv\n∈\n\n∂\n\n−\nB\n\n\n\n\n\n|\n\n\nG\nB\n\n\n(\n\nu\n,\nv\n,\nE\n\n)\n\n\n|\n\n<\n1\n\n\n, we have for the function \n\n\nf\n:\nx\n↦\n\n|\n\n\nG\nΛ\n\n\n(\n\nx\n,\ny\n,\nE\n\n)\n\n\n|\n\n\n\n a subharmonic‐type inequality:
As long as all points \n\n\nv\n\'\n\n\n at distance \n\n\nL\n+\n1\n\n\n from \n\nx\n\n are centers of \n\nL\n\n‐balls with the same “subharmonic” property, the GRI can be iterated. If \n\nn\n\n steps of iteration can be performed, and \n\n\n\n\n\n‖\nf\n‖\n\n\n∞\n\n≤\nM\n\n\n for some \n\n\nM\n<\n∞\n\n\n, then the value \n\n\nf\n\n(\nx\n)\n\n\n\n admits a small upper bound by \n\n\nM\n\nq\nn\n\n\n\n.
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Definition 3. Given real numbers \n\nE\n\n and \n\n\nm\n>\n0\n\n\n, a ball \n\n\nB\n=\nB\n\n(\n\nL\n,\nx\n\n)\n\n\n\n is called \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐nonsingular (\n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐, NS, in short), if for all \n\n\ny\n∈\n\n∂\n−\n\nB\n\n\n
with \n\n\na\n(\nm\n,\nL\n)\n:\n=\nm\n(\n1\n+\n\nL\n\n\n\n−\n1\n\n8\n\n\n\n)\n\n\n. Otherwise, it is called \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐singular (\n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐S).
Theorem 2. There exist \n\n\nδ\n>\n0\n\n\n, an interval \n\n\nI\n=\n\n[\n\n0\n,\n\nE\n*\n\n\n]\n\n\n\n with \n\n\n\nE\n*\n\n>\n0\n\n\n, and an integer \n\n\n\nL\n*\n\n\n\n such that for all \n\n\nE\n∈\nI\n\n\n and \n\n\nL\n≥\n\nL\n*\n\n\n\n one has
We shall need a positive number \n\n\nβ\n∈\n\n(\n\n0\n,\n1\n\n)\n\n\n\n; t suffices to set \n\n\nβ\n=\n1\n/\n2\n\n\n, which will be assumed below, but for clarity, sometimes the parameter \n\nβ\n\n will be used in its symbolic form.
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We denote by \n\n\nS\np\n\n(\n\n\nH\n\nB\n,\nV\n\n\n\n)\n\n\n\n the spectrum of the operator \n\n\n\nH\n\nB\n,\nV\n\n\n\n\n.
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Definition 4. Given an operator \n\n\n\nH\n\nB\n,\nV\n\n\n=\n−\n\nΔ\nB\n\n+\nV\n\n\n in a ball \n\n\nB\n=\nB\n\n(\n\nL\n,\nx\n\n)\n\n\n\n, this ball is called \n\nE\n\n‐nonresonant (\n\nE\n\n‐NR, in short), if \n\n\ndist\n\n(\n\nS\np\n\n(\n\n\nH\n\nB\n,\nV\n\n\n\n)\n\n\n)\n\n,\nE\n≥\n\n\n\n\n\n\ne\n\n−\n\nL\nβ\n\n\n\n\n\n, and \n\nE\n\n‐resonant (\n\nE\n\n‐R), otherwise.
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Clearly, if a ball is \n\nE\n\n‐NR, the resolvent is well‐defined at the energy \n\nE\n\n, and the modulus of all the respective Green functions are upper bounded by \n\n\n\ne\n\n\nL\nβ\n\n\n\n\n\n, since the finite‐dimensional operator \n\n\n\nH\n\nB\n,\nV\n\n\n\n\n is self‐adjoint. Probabilistic bounds on \n\n\ndist\n\n(\n\nS\np\n\n(\n\n\nH\n\nB\n,\nV\n\n\n\n)\n\n\n)\n\n,\nE\n\n\n for random operators are traditionally called Wegner bounds, due to the original work by Wegner [15] who established the first general bound of that kind.
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Lemma 3 (Wegner estimate). Assume that the random potential of the operator \n\n\n\nH\n\nB\n,\nV\n\n\n\n(\nω\n)\n\n=\n−\n\nΔ\nB\n\n+\nV\n\n(\nω\n)\n\n\n\n is i.i.d. and the common marginal probability distribution of \n\n\nV\n\n(\n\nx\n,\nω\n\n)\n\n\n\n admits a probability density bounded by some \n\n\n\nC\nW\n\n<\n∞\n\n\n. Then for any \n\n\ns\n∈\n\n[\n\n0\n,\n1\n\n]\n\n\n\n
Definition 5. Given an operator \n\n\n\nH\n\nB\n,\nV\n\n\n=\n−\n\nΔ\nB\n\n+\nV\n\n\n in a ball \n\n\nB\n=\nB\n\n(\n\n\nL\n\nk\n+\n1\n\n\n,\nx\n\n)\n\n\n\n, this ball is called \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐bad if it contains at least two nonoverlapping \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐S balls of radius \n\n\n\nL\nk\n\n\n\n, and \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐good, otherwise.
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Lemma 4. If a ball \n\n\nB\n\n(\n\n\nL\n\nk\n+\n1\n\n\n,\nx\n\n)\n\n\n\n is \n\nE\n\n‐NR and \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐good, then it is \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐NS.
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Sketch of the proof. The claim is easily obtained by iterating the Simon‐Lieb inequality (SLI) and using the hypothesis of \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐goodness; the latter guarantees that in the course of iterated applications of the GRI, one can stumble on an \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐S ball of size \n\n\n\nL\nk\n\n\n\n at most once. There may be no singular ball inside \n\n\nB\n\n(\n\n\nL\n\nk\n+\n1\n\n\n,\nx\n\n)\n\n\n\n, then the subharmonic‐type inequalities easily provide an exponential decay from the center to the boundary of the ball. Furthermore, if there is one singular \n\n\n\nL\nk\n\n\n\n ball, one can approach it from the center and from the boundary, using the SLI on the first or on the second spatial argument of the Green function. The “wasted” distance is of order or \n\n\nO\n\n(\n\n\nL\nk\n\n\n)\n\n\n\n, so an elementary calculation provides the desired decay bound of the Green’s functions. Technical details can be found in Ref. [16], but it is worth emphasizing the crucial role of the “nonresonance” hypothesis: as was explained, an iterated use of the subharmonic‐type inequality in Eq. (21) only gives the upper bound \n\n\nf\n\n(\nx\n)\n\n≤\n\nq\nn\n\n\n\n\n‖\nf\n‖\n\n\n∞\n\n\n\n, which is absolutely useless without an explicit control of the sup‐norm of the function \n\nf\n\n, and in our case, one has the functions \n\n\nf\n:\nx\n↦\n\n|\n\n\nG\nΛ\n\n\n(\n\nx\n,\ny\n,\nE\n\n)\n\n\n|\n\n\n\n.
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Theorem 2 can be derived from the following inductive statement.
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Theorem 3. Introduce the following notations: for each \n\n\nk\n≥\n0\n\n\n, let
Then for all \n\n\nk\n≥\n0\n\n\n one has \n\n\n\nP\nk\n\n≤\n\ne\n\n−\n\nL\nk\n\n\n\n\nδ\n\n\n\n\n\n.
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Sketch of the proof. We proceed by induction, starting with the hypothesis \n\n\n\nP\n0\n\n≤\n\ne\n\n−\n\nL\nk\n\n\n\n\nδ\n\n\n\n\n\n. Assume the required bound holds for some \n\n\nk\n≥\n0\n\n\n, then we have to prove it for the balls of size \n\n\n\nL\n\nk\n+\n1\n\n\n\n\n. By Lemma 4, if a ball \n\n\nB\n\n(\n\n\nL\n\nk\n+\n1\n\n\n,\nx\n\n)\n\n\n\n is \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐S, then either it is \n\nE\n\n‐R, or it is not \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐good, i.e., contains at least two disjoint balls \n\n\nB\n\n(\n\n\nL\nk\n\n,\nu\n\'\n\n)\n\n\n\n, \n\n\nB\n\n(\n\n\nL\nk\n\n,\nu\n\'\n\'\n\n)\n\n\n\n which are \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐S. The number of possible pairs \n\n\n\n(\n\n\nu\n\'\n\n,\n\nu\n\n\'\n\'\n\n\n\n)\n\n\n\n inside \n\n\nB\n\n(\n\n\nL\n\nk\n+\n1\n\n\n,\nx\n\n)\n\n\n\n is bounded by \n\n\n\n1\n2\n\n\nC\nd\n2\n\n\nL\n\nk\n+\n1\n\n2\n\n\n\n, and the probability for each pair \n\n\nB\n\n(\n\n\nL\nk\n\n,\n\nu\n\'\n\n\n)\n\n,\nB\n\n(\n\n\nL\nk\n\n,\nu\n\'\n\'\n\n)\n\n\n\n to be \n\n\n\n(\n\nE\n,\nm\n\n)\n\n\n\n‐S is bounded inductively by \n\n\n\nP\nk\n\n≤\n\ne\n\n−\n\nL\nk\n\n\n\n\nδ\n\n\n\n\n\n. Thus, the probability of existence of at least one such pair is upper‐bounded by
Further, the probability for the ball \n\n\nB\n\n(\n\n\nL\n\nk\n+\n1\n\n\n,\nx\n\n)\n\n\n\n to be \n\nE\n\n‐R is upper‐bounded with the help of the Wegner estimate from Lemma 111, without induction:
Now the claim follows by a straightforward, albeit somewhat cumbersome calculation, making use of the assumed geometrical condition in Eq. (29). The details can be found in Ref. [16].
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Theorem 3 shows that if on some scale \n\n\n\nL\n0\n\n\n\n the Green functions in the balls of radius \n\n\n\nL\n0\n\n\n\n decay—with a sufficiently high probability—exponentially fast from the center to the boundary of the ball, then the same phenomenon is reproduced, with ever higher probability, on any scale \n\n\n\nL\nk\n\n→\n+\n∞\n\n\n. Such a decay is akin to that of a wave function of a quantum particle in a classically prohibited space where the energy of the particle is below the potential “barrier,” so there is a powerful mechanism, originally discovered by P. W. Anderson in 1958, which reproduces the local tendency of a quantum particle to localization in the disordered environment on any scale. The main problem concerns the mechanisms creating such a tendency for localization. This is where we turn to the spectral analysis in the balls \n\n\nB\n\n(\n\n\nL\n0\n\n,\nx\n\n)\n\n\n\n and seek estimates for the first nonzero eigenvalue of the graph Laplacian in \n\n\nB\n\n(\n\n\nL\n0\n\n,\nx\n\n)\n\n\n\n.
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Indeed, if we restrict our analysis to the interval of small positive energies (assuming the potential is nonnegative; otherwise we can make a spectral shift), then it is clear that for all energies \n\nE\n\n below the spectrum of the operator \n\n\n\nG\n\nB\n,\nV\n\n\n\n(\n\nx\n,\ny\n,\nE\n\n)\n\n\n\n must decay exponentially with respect to the distance \n\n\n\n|\n\nx\n−\ny\n\n|\n\n\n\n, due to the above‐mentioned “under‐the‐barrier” decay well‐known from the elementary exercises in quantum mechanics. Mathematically, there is actually a more general result, the Combes‐Thomas estimate [17] which applies not only to the values \n\nE\n\n strictly below the spectrum of \n\n\n\nH\n\nB\n,\nV\n\n\n\n\n, but all \n\nE\n\n in the resolvent set, i.e., simply away from the spectrum. Specifically, fix an operator \n\n\n\nH\n\nB\n,\nV\n\n\n\n\n relative to a ball \n\n\nB\n\n(\n\nL\n,\nu\n\n)\n\n\n\n, and let \n\n\nE\n∈\nR\n\n\n satisfy \n\n\ndist\n\n(\n\nE\n,\nS\np\n\n(\n\n\nH\n\nB\n,\nV\n\n\n\n)\n\n≥\nη\n\n)\n\n\n\n. There exist universal constants \n\n\nC\n,\n\nC\n\'\n\n>\n0\n\n\n such that for all \n\n\nx\n,\ny\n∈\nB\n\n(\n\nL\n,\nu\n\n)\n\n\n\n it holds that
Now all the pieces of the puzzle find their place:
Using the isoperimetric spectral estimates combined with the Temple inequality, we can find a sufficiently small energy interval \n\n\n\n[\n\nE\n,\n\nE\n*\n\n+\nη\n\n]\n\n\n\n, with \n\n\n\nE\n*\n\n,\nη\n>\n0\n\n\n, such that in large balls \n\n\nB\n\n(\n\n\nL\n0\n\n,\nu\n\n)\n\n\n\n a random potential takes a very low average value with a very small probability, so that it is highly unlikely for \n\n\n\nH\n\nB\n,\nV\n\n\n\n(\nω\n)\n\n\n\n to have its lowest eigenvalue below \n\n\n\nE\n*\n\n+\nη\n\n\n.
Restrict the energy interval to \n\n\n\nI\n*\n\n:\n=\n\n[\n\nE\n,\n\nE\n*\n\n\n]\n\n\n\n; then for all \n\n\nE\n∈\n\nI\n*\n\n\n\n we have \n\n\ndist\n\n(\n\nE\n,\nS\np\n\n(\n\n\nH\n\nB\n,\nV\n\n\n\n)\n\n≥\nη\n\n)\n\n\n\n, so the Combes‐Thomas estimate in Eq. (31) applies and guarantees a fast decay of the Green functions from the center to the boundary of the ball \n\n\nB\n\n(\n\nL\n,\n\n)\n\n\n\n. Notice that we can have \n\nη\n\n lower bounded by a fractional power of \n\nL\n\n. Indeed, Eq. (19) allows us to take \n\n\nη\n=\n\nL\n\n−\n1\n/\n4\n\n\n\n\n, and the probability for such a bound to hold is at least \n\n\n1\n−\n\ne\n\n−\nδ\n/\n12\n\n\n\n\n, in notations of Eq. (19).
We thus have, in a tiny interval of energies close to the bottom of the spectrum, the starting hypothesis of the scale induction fulfilled. Now the roll the induction and prove exponential decay with high probability at any scale \n\n\n\nL\nk\n\n\n\n, \n\n\nk\n≥\n0\n\n\n.
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Once again, it is to be stressed that it is a graph‐theoretic spectral estimate that makes this story possible, and the presented phenomena take place for a rich class of graphs, much larger than just periodic lattices. This general estimate is a far‐going replacement for the elementary consequences of the Fourier analysis on \n\n\n\nZ\nd\n\n\n\n.
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Summarizing, the problem of computing exact asymptotics, or at least sharp upper/lower bounds on the limiting distribution function of the eigenvalues for the operators\n\n\n\nH\n\nB\n,\nV\n\n\n\n(\nω\n)\n\n\n\non various classes of graphs is of course much more general and important than one particular application to the Anderson localization presented above. This if often a difficult problem, and the wealth of knowledge and intuition accumulated in the spectral graph theory would be very welcome to this area of mathematical physics.
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3. Symmetric powers of graphs and spectra of fermionic systems
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3.1. Motivation and preliminaries
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Now we turn to another problem of spectral analysis of quantum Hamiltonians of disordered systems. The presentation will be less technical, and the main message is that the graph theory provides here both an adequate language and technical tools allowing one to treat efficiently difficult problems arising in the recently developed multiparticle localization theory; some of these problems are still open and challenging.
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In quantum mechanics, stationary states of a system of several quantum particles are described by the eigenfunctions of their respective Hamiltonians acting in subspaces of either symmetric or antisymmetric functions \n\n\n\nΨ\n\n\n(\nx\n)\n\n\n\n, \n\n\nx\n=\n\n(\n\n\nx\n1\n\n,\n…\n,\n\nx\nN\n\n\n)\n\n\n\n, where \n\n\nN\n≥\n1\n\n\n is the number of particles, and each argument \n\n\n\nx\nj\n\n\n\n runs through the single‐particle configuration space. The particles described by antisymmetric functions are called fermions, and those described by symmetric functions are called bosons. Physically speaking, the particles evolve in the three‐dimensional space, but in the framework of the so‐called tight‐binding approximation, they can be restricted to a periodic lattice or, more generally, a locally finite graph embedded in the Euclidean space. In this section, we assume the latter and work with N‐particle systems on a graph \n\nΓ\n\n. In fact, even the case where \n\n\nΓ\n=\n\nZ\nd\n\n\n\n is of interest for us, since we are going now to show how a typical construction of the graph theory, the symmetric power of a graph, can be instrumental for solving a formal yet thorny technical problem encountered in the multiparticle Anderson localization theory.
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The quantum particles are physically indistinguishable, so any accurate mathematical model has to reflect this fact. In some situations including the localization analysis of randomly disordered media, it is more convenient to represent the Hilbert space of symmetric or antisymmetric functions on \n\nΓ\n\n as the space of functions on the set of configurations of \n\nN\n\n indistinguishable particles, instead of a subspace of (+/−)‐symmetric functions defined directly on \n\nΓ\n\n. While the two approaches are mathematically equivalent, the latter one has an important technical advantage that can be explained as follows.
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Consider for simplicity of a two‐particle fermionic system in a finite subgraph \n\n\nG\n⊂\n\n[\n\n0\n,\nL\n\n]\n\n⊂\nZ\n\n\n with the graph structure inherited from \n\nZ\n\n. The wave functions of the two‐particle systems are thus antisymmetric functions \n\n\n\nΨ\n\n\n(\nx\n)\n\n=\n\nΨ\n\n\n(\n\n\nx\n1\n\n,\n\nx\n2\n\n\n)\n\n\n\n of two variables \n\n\n\nx\n1\n\n,\n\nx\n2\n\n∈\nG\n\n\n. We assume the Hamitlonian of this system to ba a discrete Schrödinger operator of the form
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Definition 6.Let be given a random potential\n\n\nV\n\n(\nω\n)\n\n\n\non a subgraph\n\nG\n\nof the lattice\n\nZ\n\n, and a nonrandom function\n\n\nr\n↦\n\nU\n\n\n(\n2\n)\n\n\n\n\n(\nr\n)\n\n\n\nof an integer argument. A two‐particle discrete Schrödinger operator on\n\n\nG\n\n\n\nis the operator of the form
\n\n\n\n\n\nH\n\n0\n\n=\n−\n\nΔ\nG\n\n\n(\n1\n)\n\n\n\n−\n\nΔ\nG\n\n\n(\n1\n)\n\n\n\n\n\n(the kinetic energy operator) is the sum of two replicas\n\n\n\nΔ\nG\n\n\n(\nj\n)\n\n\n\n\n\nof the graph Laplacian on\n\nG\n\n, acting on a function\n\n\n\nΨ\n\n\n(\n\n\nx\n1\n\n,\n\nx\n2\n\n\n)\n\n\n\n as a function of the variable\n\n\n\nx\nj\n\n\n\n,\n\n\nj\n=\n1\n,\n2\n\n\n;
\n\n\n\n\nV\n\n\n(\n\nx\n,\nω\n\n)\n\n\n\nis the operator of multiplication by the random function\n\n\n\n(\n\n\nx\n1\n\n,\n\nx\n2\n\n\n)\n\n↦\n\nV\n\n\n(\n\n\nx\n1\n\n,\nω\n\n)\n\n+\n\nV\n\n\n(\n\n\nx\n2\n\n,\nω\n\n)\n\n\n\n; and
\n\n\n\n\nU\n\n\n(\nx\n)\n\n\n\nis the operator of the interaction energy of the two particles at hand, acting as the operator of multiplication by the nonrandom function\n\n\n\n(\n\n\nx\n1\n\n,\n\nx\n2\n\n\n)\n\n↦\n\n\nU\n\n\n\n(\n2\n)\n\n\n\n\n(\n\n\n|\n\n\nx\n1\n\n−\n\nx\n2\n\n\n|\n\n\n)\n\n\n\n.
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The factor \n\n\nϵ\n>\n0\n\n\n measures the amplitude of the kinetic energy operators and reflects the mobility of the particles. In this section, it is instructive to think of \n\nϵ\n\n as a small number, so that the potential energy is in a certain sense dominant. \n\n\n\nV\n\n\n(\n\n\nx\n\n,\nω\n\n)\n\n\n\n is the operator of random potential energy induced by the disordered media (modeled here by a finite linear chain) acting as operator of multiplication by the real‐valued function
and \n\n\n\nV\n\n\n(\n\nx\n,\nω\n\n)\n\n\n\n are i.i.d. random variables on \n\nG\n\n (local potentials produced, e.g., by heavy ions).
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The function \n\n\n\n\nU\n\n\n\n(\n2\n)\n\n\n\n\n(\nr\n)\n\n\n\n is called the two‐body interaction potential.
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For the sake of notational clarity, here and below we use boldface notations for various objects related to multiparticle objects.
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The onset of Anderson localization manifests itself by a fast (usually exponential) decay of the eigenstates \n\n\n\n\nΨ\n\nk\n\n\n\n of the Hamiltonian \n\n\n\nH\n\n\n(\nω\n)\n\n\n\n away from some vertex, depending upon the quantum number $j$ (usually referred to as the localization center of the respective eigenstate \n\n\n\n\nΨ\n\nk\n\n\n\n. The quantum transport, on the other hand, may take place due to the tunneling between distant vertices \n\n\nx\n,\n\ny\n\n∈\nG\n×\nG\n\n\n with very close local energies. The latter notion can be ambiguous when the kinetic energy is nonnegligent, but pictorially, under the assumption we made above that \n\n\nϵ\n\n\n is small, the local energy at a vertex \n\nx\n\n is essentially given by \n\n\n\nV\n\n\n(\n\n\nx\n\n,\nω\n\n)\n\n+\n\nU\n\n\n(\n\nx\n\n)\n\n\n\n, thus depends directly upon \n\nx\n\n.
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Now recall that the modulus of an asymmetric function \n\n\n\nΨ\n\n\n(\n\n\nx\n1\n\n,\n\nx\n2\n\n\n)\n\n\n\n is symmetric, thus \n\n\n|\n\nΨ\n\n\n(\n\nx\n\n)\n\n|\n\n\n necessarily takes identical values at vertices \n\n\nx\n=\n\n(\n\n\nx\n1\n\n,\n\nx\n2\n\n\n)\n\n\n\n and \n\n\nS\nx\n=\n\n(\n\n\nx\n2\n\n,\n\nx\n1\n\n\n)\n\n\n\n in the space of ordered configurations of distinguishable particles. The symmetric vertices \n\nx\n\n and \n\n\nS\nx\n\n\n can be located at arbitrarily large distances from each other in the original two‐particle configuration space given by the Cartesian product \n\n\nG\n×\nG\n\n\n. As a result, one has, formally, consider the possibility of “tunneling” between \n\n\nx\n\n\n and \n\n\nS\nx\n\n\n, although there is no physical particle transfer process between these two configurations: from the consistent quantum mechanical point of view, the latter are simply identical! We come therefore to realize that the mathematical model based on the Cartesian square of the “physical,” single‐particle configuration space \n\nG\n\n generates some formal problems which actually have no physical raison d\'être, yet they have to be addressed explicitly to rule out some unwanted phenomena. In particular, this renders substantially more complicated the rigorous localization analysis.
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However, the above‐mentioned difficulty disappears as soon as one replaces the Hilbert space of antisymmetric functions \n\n\n\nΨ\n\n\n(\n\n\nx\n1\n\n,\n\nx\n2\n\n\n)\n\n\n\n on \n\n\nG\n×\nG\n\n\n by an isomorphic Hilbert space of functions Φ on the set of configurations of indistinguishable pairs of vertices from the basic graph \n\nG\n\n. The required construction is well‐known in the graph theory: we need a symmetric power \n\n\n\nG\n\n\n(\n2\n)\n\n\n\n\n\n of the graph \n\nG\n\n. Due to the mathematical complexity of the rigorous multiparticle Anderson localization theory, we can only sketch its general strategy in this chapter, but the main tool, which proves very valuable here, deserves a more detailed discussion. As was said in the introductory part, the main goal of this section is to attract the readers’ attention to some interesting and useful relations between the mathematical physics of disordered media and the notions, tools, and deep results of the graph theory.
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3.2. Construction of a symmetric power of a graph
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3.2.1. An example in one dimension
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Before we turn to general constructions of symmetric powers of locally finite graphs, it seems instructive to consider first a particular case where the underlying, basic graph \n\nG\n\n is linear, i.e., isomorphic to a subgraph of the one‐dimensional lattice \n\nZ\n\n. The existence of a complete linear order makes possible a particularly simple variant of the symmetric square \n\n\n\nG\n\n\n(\n2\n)\n\n\n\n\n\n (indeed, of any symmetric power \n\n\n\nG\n\n\n(\nN\n)\n\n\n\n\n\n, \n\n\nN\n>\n1\n\n\n).
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Consider the triangular subset of lattice square \n\n\nQ\n(\nL\n)\n:\n=\nG\n×\nG\n=\n\n〚\n\n0\n,\nL\n\n〛\n\n×\n\n〚\n\n0\n,\nL\n\n〛\n\n\n\n (here \n\n\nG\n=\n\n〚\n\n0\n,\nL\n\n〛\n\n\n\n stands for the integer interval \n\n\n[\n0\n,\nL\n]\n∩\nZ\n\n\n):
(here (2) reflects the number of “particles”). An example is presented in Figure 1. Any antisymmetric function \n\n\nΨ\n\n\n on \n\n\nQ\n\n(\nL\n)\n\n\n\n vanishes at any point of the form \n\n\n\n(\n\nx\n,\nx\n\n)\n\n\n\n, and its modulus takes identical values on \n\n\n\n(\n\n\nx\n1\n\n,\n\nx\n2\n\n\n)\n\n∈\n\nG\n\n\n(\n2\n)\n\n\n\n\n\n and on the symmetric point \n\n\n\n(\n\n\nx\n2\n\n,\n\nx\n1\n\n\n)\n\n\n\n. It follows that
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Figure 1.
Example of a symmetric square. Here the base graph \n\nG\n\n is a subgraph of \n\nZ\n\n, and it can be implemented as a subgraph of the Cartesian square \n\n\n\nZ\n2\n\n\n\n, owing to the one‐dimensional topology of \n\nZ\n\n.
This provides a natural isomorphism between the Hilbert spaces of complex functions on \n\n\nQ\n\n(\nL\n)\n\n\n\n and on \n\n\n\nG\n\n\n(\n2\n)\n\n\n\n.\n\n\n Clearly, the same idea works in the N‐particle case, where we can define
except that in the latter case, the factor of \n\n\n2\n=\n2\n!\n\n\n in Eq. (23) is to be replaced by \n\n\nN\n!\n\n\n.
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Such a simple, transparent geometrical construction is no longer available for general graphs, but an isomorphism similar to that from Eq. (23) can be established for the symmetric powers of graphs. Below we give a variant with a distinctive flavor of quantum mechanics.
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3.2.2. General construction
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The vertex set. Let be given a connected, locally finite countable graph with the vertex set \n\nG\n\n and an edge set \n\nE\n\n, and an integer \n\n\nN\n≥\n2\n\n\n. Consider the integer‐valued functions \n\n\n\nn\n\n:\nx\n↦\n\nn\n\n\n(\nx\n)\n\n∈\n\n{\n\n0\n,\n1\n\n}\n\n\n\n on \n\nG\n\n such that \n\n\n\n\n∑\n\n\nx\n∈\nG\n\n\n\nn\n\n\n(\nx\n)\n\n=\nN\n\n\n. The physical meaning of the value \n\n\n\nn\n\n\n(\nx\n)\n\n\n\n is the number of particles at the vertex \n\nx\n\n, so it is usually called the occupation number of the site \n\n\nx\n∈\nG\n\n\n. Due to the indistinguishable nature of the particles, only the numbers of particles at each site are physically observable (measurable in experiments). Furthermore, since we are modeling now fermions, the respective wave functions, by their antisymmetry, must vanish on any configuration of \n\nN\n\n particles among which at least two occupy the same position. This was precisely the reason we excluded the “diagonal” from \n\n\n\nG\n\n\n(\n2\n)\n\n\n\n\n\n above. Now we achieve the same effect by the requirement \n\n\n\nn\n\n\n(\nx\n)\n\n∈\n\n{\n\n0\n,\n1\n\n}\n\n\n\n. The bottom line is that a configuration of \n\nN\n\n particles admissible for modeling fermions is completely determined by a function \n\n\nn\n\n\n; for all intents and purposes, each \n\n\nn\n\n\n is a (fermionic) configuration.
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Hence, we constructed an appropriate vertex set \n\n\n\nG\n\n\n(\nN\n)\n\n\n\n\n\n of the graph that would generalise \n\n\n\nG\n\n\n(\n2\n)\n\n\n\n\n\n for an general underlying graph \n\nG\n\n. Specifically, there is a projection
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\n\n\n\n\nΠ\n\n\n(\nN\n)\n\n\n\n:\n\nx\n\n=\n\n(\n\n\nx\n1\n\n,\n…\n,\n\nx\nN\n\n\n)\n\n↦\n\n\nn\n\nx\n\n,\n where \ns\nu\np\np\n\n\n\nn\n\nx\n\n=\n\n{\n\n\nx\n1\n\n,\n…\n,\n\nx\nN\n\n\n}\n\n.\n\n\n\n\nE31
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The points of the support of a function \n\n\n\n\nn\n\nx\n\n\n\n will be called the particles of the configuration\n\n\n\n\nn\n\nx\n\n\n\n. Restricted on the set of \n\nN\n\n‐tuples of pairwise distinct vertices \n\n\n\nx\n1\n\n,\n…\n,\n\nx\nN\n\n\n\n, the projection \n\n\n\nΠ\n\n\n(\nN\n)\n\n\n\n\n\n is exactly \n\n\nN\n!\n\n\n‐fold: the pre‐image \n\n\n\n\n\n(\n\n\nΠ\n\n\n(\nN\n)\n\n\n\n\n)\n\n\n\n−\n1\n\n\n\n(\n\nn\n\n)\n\n\n\n has cardinality \n\n\nN\n!\n\n\n.
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The edge set. Using again a terminology inspired by physics, we say that two configurations \n\n\n\n\nn\n\n′\n\n,\n\n\n\nn\n\n′\n\n′\n\n\n\n form an (unordered) edge if and only if \n\n\n\n\nn\n\n′\n\n′\n\n\n is obtained by moving exactly one particle of \n\n\n\nn\n\n′\n\n\n to a position unoccupied by other particles from \n\n\n\nn\n\n′\n\n\n.
It is not difficult to see that the two definitions are equivalent. Indeed, each term in the above sum equals \n\n0\n\n or \n\n1\n\n, since \n\n\n0\n≤\n\n\nn\n\n′\n\n\n(\nx\n)\n\n,\n\n\nn\n\n″\n\n\n(\nx\n)\n\n≤\n1\n\n\n, and such a term vanishes when either \n\n\n\n\nn\n\n′\n\n\n(\nx\n)\n\n=\n\n\nn\n\n″\n\n\n(\nx\n)\n\n=\n0\n\n\n, i.e., \n\nx\n\n is unoccupied by either configuration, or \n\n\n\n\nn\n\n′\n\n\n(\nx\n)\n\n=\n\n\nn\n\n″\n\n\n(\nx\n)\n\n=\n1\n\n\n, i.e., both configurations have a particle at \n\nx\n\n. Removing particles from the support of \n\n\n\nn\n\n′\n\n\n to produce new configuration \n\n\n\n\n\nn\n\n′\n\n′\n\n\n\n, we have to place them outside the support. Therefore,
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each point \n\nx\n\n with \n\n\n\n\nn\n\n′\n\n\n(\nx\n)\n\n=\n1\n\n\n and \n\n\n\n\nn\n\n″\n\n\n(\nx\n)\n\n=\n0\n\n\n (i.e., occupied by \n\n\n\nn\n\n′\n\n\n but unoccupied by \n\n\n\n\n\nn\n\n′\n\n′\n\n\n\n) contributes by a two unit term to the sum: first, \n\n\n\n\nn\n\n′\n\n\n(\nx\n)\n\n−\n\n\nn\n\n″\n\n\n(\nx\n)\n\n=\n1\n\n\n, and second, for the position \n\ny\n\n to which we move the particle from \n\nx\n\n we have \n\n\n\n\nn\n\n′\n\n\n(\nx\n)\n\n−\n\n\nn\n\n″\n\n\n(\nx\n)\n\n=\n−\n1\n\n\n. If we move more than one particle from \n\n\n\nn\n\n′\n\n\n, the sum in Eq. (25) will be at least \n\n4\n\n. We conclude that the second definition in Eq. (25) is equivalent to the first one.
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This completes the construction of the \n\nN\n\n‐th symmetric power \n\n\n(\n\nG\n\n(\nN\n)\n\n\n,\n\nE\n\n(\nN\n)\n\n\n)\n\n\n of the graph \n\n\n\n(\n\nG\n,\nE\n\n)\n\n\n\n.
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3.2.3. Hilbert space isomorphism: antisymmetric functions versus symmetric power
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Now the isomorphism between of the Hilbert space of square‐summable complex antisymmetric functions on the Cartesian power \n\n\n\nG\nN\n\n\n\n and the Hilbert space of all square‐summable complex functions on \n\n\n\nG\n\n\n(\nN\n)\n\n\n\n\n\n is defined in a natural fashion. Recall that we introduced the projection \n\n\n\nΠ\n\n\n(\nN\n)\n\n\n\n\n\n, such that \n\n\n\n\n\n(\n\n\nΠ\n\n\n(\nN\n)\n\n\n\n\n)\n\n\n\n−\n1\n\n\n\n(\n\n\n\nn\n\nx\n\n\n)\n\n\n\n consists of all \n\n\nN\n!\n\n\n permutations of the vertices \n\n\n\nx\n1\n\n,\n…\n,\n\nx\nN\n\n\n\n from the support of \n\n\n\n\nn\n\nx\n\n\n\n. Any function Φ\n\n\n:\n\nG\n\n\n(\nN\n)\n\n\n\n→\nC\n\n\n generates a symmetric function \n\n\n\nΨ\n\n=\n\nΦ\n\n∘\n\nΠ\n\n\n(\nN\n)\n\n\n\n\n\n, and each symmetric function \n\n\n\nΨ\n\n:\n\nG\nN\n\n→\nC\n\n\n can be obtained in this way, as \n\n\n\nΦ\n\n∘\n\nΠ\n\n\n(\nN\n)\n\n\n\n\n\n. The modulus of any antisymmetric function \n\n\nΨ\n\n\n on the Cartesian power \n\n\n\nG\nN\n\n\n\n takes identical values on all elements of \n\n\n\n\n\n(\n\n\nΠ\n\n\n(\nN\n)\n\n\n\n\n)\n\n\n\n−\n1\n\n\n\n(\n\n\n\nn\n\nx\n\n\n)\n\n\n\n. Thus, with \n\n\n\n\nΨ\n\n\nΦ\n\n\n:\n=\n\nΦ\n\n∘\n\nΠ\n\n\n(\nN\n)\n\n\n\n\n\n,
Now the required isomorphism is defined by \n\n\n\n\n\n‖\n\n\n\nΨ\n\n\nΦ\n\n\n\n‖\n\n\n2\n\n=\n\n\nN\n!\n\n\n\n\n\n‖\n\nΦ\n\n‖\n\n\n2\n\n\n\n.
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It might appear that the described isomorphism is just an interesting formal trick, but there is much more to it than a mere mathematical curiosity. As one illustration, we consider now a problem of spectral analysis for multiparticle random Hamiltonians \n\n\n\nH\n\n\n(\nω\n)\n\n\n\n above.
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3.2.4. An application: KAM‐type analysis of two‐particle fermionic random Hamiltonians
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The goal of this subsection is merely to illustrate the key role of the isomorphism between the subspace of antisymmetric square‐summable functions of \n\n\nN\n>\n1\n\n\n variables in a graph \n\nG\n\n and the space of all square‐summable functions on the symmetric power \n\n\n\nG\n\n\n(\nN\n)\n\n\n\n\n\n. The mathematical problem where it is used is quite complex, so we will only sketch the main setting and focus on the role of the symmetric powers.
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It is to be emphasized that the spectral analysis of \n\nN\n\n‐particle quantum Hamiltonians in presence of a random potential field, generated by a disordered media, accurately taking into account a nontrivial interaction between the particles, is a relatively new direction both in theoretical physics and in rigorous mathematical physics. This is an actively developing and challenging area of research. While physicists have finally obtained convincing theoretical results on the stability of localization under the Coulomb interaction, the progress in rigorous mathematical physics is still relatively modest, compared to theoretical physics. The best insight achieved in the last 10 years concerns the systems of a fixed number \n\nN\n\n of particles in a large sample of a disordered media. For the purposes of this chapter, we always restrict the analysis to discrete systems on graphs.
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As was said in the introductory section, we can only sketch rather complex mathematical constructions involved and illustrate the main mechanisms of stability of localization under interaction. For simplicity, suppose that we have a system of \n\n\nN\n=\n2\n\n\n particles in a large but finite connected graph \n\nG\n\n on which an i.i.d. random (potential) field \n\n\nx\n↦\n\nV\n\n\n(\n\nx\n,\nω\n\n)\n\n\n\n is defined. One can consider various marginal probability distributions, i.e., identical probability distributions of the random variables \n\n\n\nV\n\n\n(\n\nx\n,\nω\n\n)\n\n,\nx\n∈\nG\n\n\n. Two models popular in theoretical physics of disordered media suit perfectly our needs here: a standard Gaussian distribution \n\n\nN\n\n(\n\n0\n,\n1\n\n)\n\n\n\n with zero mean and unit variance, and the uniform distribution on the interval \n\n\n\n[\n\n0\n,\n1\n\n]\n\n\n\n.
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Consider first the simplest (yet mathematically nontrivial) case of zero amplitude of interaction. Then the variables in the Hamiltonian \n\n\n\nH\n\n\n(\nω\n)\n\n\n\n can be separated, since in this case one has an algebraic representation \n\n\n\nH\n\n\n(\nω\n)\n\n=\n\n\nH\n\n\n\n(\n1\n)\n\n\n\n\n(\nω\n)\n\n⊗\nI\n+\nI\n⊗\n\n\nH\n\n\n\n(\n2\n)\n\n\n\n\n(\nω\n)\n\n\n\n where \n\nI\n\n is the identity operator, and therefore, the eigenvectors of the operator \n\n\n\nH\n\n\n(\nω\n)\n\n\n\n can be chosen in the tensor product form, \n\n\n\n\nΨ\n\n\ni\n,\nj\n\n\n\n(\nω\n)\n\n=\n\nψ\ni\n\n\n(\nω\n)\n\n⊗\n\nψ\nj\n\n\n(\nω\n)\n\n\n\n, where \n\n\n\nψ\ni\n\n\n(\nω\n)\n\n,\n\nψ\nj\n\n\n(\nω\n)\n\n\n\n are eigenvectors of the single‐particle Hamiltonians \n\n\n\n\nH\n\n\n\n(\n1\n)\n\n\n\n\n(\nω\n)\n\n\n\n and \n\n\n\n\nH\n\n\n\n(\n2\n)\n\n\n\n\n(\nω\n)\n\n\n\n, respectively. The latter are identical replicas acting on their respective variables \n\n\n\nx\n1\n\n\n\n and \n\n\n\nx\n2\n\n\n\n. The eigenvalues are the sums \n\n\n\nE\n\ni\n,\nj\n\n\n\n(\nω\n)\n\n=\n\nE\ni\n\n\n(\n1\n)\n\n\n\n\n(\nω\n)\n\n+\n\nE\nj\n\n\n(\n2\n)\n\n\n\n\n(\nω\n)\n\n\n\n, with \n\n\n\n\nH\n\n\n\n(\n\n1\n,\n2\n\n)\n\n\n\n\n(\nω\n)\n\n\nψ\ni\n\n\n(\nω\n)\n\n=\n\nE\ni\n\n\n(\n\n1\n,\n2\n\n)\n\n\n\n\n(\nω\n)\n\n\nψ\ni\n\n\n(\nω\n)\n\n\n\n.
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The main reason why the electron‐electron interaction was consciously neglected even in theoretical physics, is that it is relatively weak as compared to the potential energy of interaction with the ions surrounding the mobile electrons, so we also assume the amplitude of the interaction potential to be small.
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Another important assumption can simplify the spectral analysis, at least in an informal treatment of the problem: small amplitude of the factor \n\nϵ\n\n in front of the kinetic energy operator \n\n\n\n\nH\n\n0\n\n\n\n; in physical terms, this corresponds to a low mobility of the particles at hand which, naturally, should favor “localization” of a given particle. Mathematically, already for \n\n\nN\n=\n1\n\n\n (isolated particles), with \n\n\nϵ\n=\n0\n\n\n we get a multiplication operator which has the orthonormal eigenbasis composed of the “discrete delta‐functions” \n\n\n\nφ\nx\n\n=\n\n\n1\n\n\n{\nx\n}\n\n\n\n\n, \n\n\nx\n∈\nG\n\n\n. Similarly, for \n\n\nN\n≥\n2\n\n\n and \n\n\nϵ\n=\n0\n\n\n we have a perator of multiplication by the total potential energy which also has a complete eigenbasis composed of “localized” eigenfunctions.
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If we had constructed an eigenbasis of the multiplication operator in the representation on the Cartesian square \n\n\n\nG\n2\n\n\n\n, then we would have obtained the two‐site supported eigenfunctions:
while the representation on the symmetric square \n\n\n\nG\n\n\n(\n2\n)\n\n\n\n\n\n gives rise to single‐site eigenfunctions \n\n\n\n\n1\n\n\n\n{\nx\n}\n\n\n\n⊗\n\n\n1\n\n\n\n{\ny\n}\n\n\n\n\n\n, where \n\n\nx\n≠\ny\n\n\n. Naïvely, starting with the “ultralocalized” eigenfunctions \n\n\n\n\n1\n\n\n\n{\nx\n}\n\n\n\n⊗\n\n\n1\n\n\n\n{\ny\n}\n\n\n\n\n\n, one may attempt to use the first‐order perturbation theory. The well‐known formulae of the rigorous perturbation theory for the eigenvectors reveal two problems:
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“small denominators,” i.e., pairs of very close or equal eigenvalues; and
large dimension of the spectral problem, which may also give rise to degenerate eigenvalues or at least to some pairs of close eigenvalues.
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Indeed, the eigenvalue associated to the unperturbed eigenfunctions of the potential energy operator \n\n\n\n\nΦ\n\n\nx\n,\ny\n\n\n=\n\n\n1\n\n\n\n{\nx\n}\n\n\n\n⊗\n\n\n1\n\n\n\n{\ny\n}\n\n\n\n\n\n is given by
Fix now the random field model with uniform marginal distribution, and let the interaction be uniformly bounded, then the above eigenvalues all belong to some fixed, bounded interval, regardless of the dimension \n\n\nD\n=\nG\n∨\n\n(\n\n\n|\nG\n|\n\n−\n1\n\n)\n\n\n\n of the Hilbert space, growing with the cardinality of \n\nG\n\n. The larger is \n\n\nG\n∨\n\n\n, the closer the \n\nD\n\n eigenvalues must get, counted with multiplicity and restricted to a fixed interval of \n\nR\n\n. In the Gaussian model, a similar phenomenon is encountered, with high probability, since large values of the Gaussian random potential \n\n\n\nV\n\n\n(\n\nx\n,\nω\n\n)\n\n\n\n are taken with small probability.
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A conclusion we can draw from this elementary analysis is that one cannot expect the perturbation theory for nondegenerate spectra, or for the finite‐dimensional operators with bounded multiplicity, to work efficiently in the model with a large graph \n\nG\n\n.
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Several approaches have been developed in spectral theory of single‐particle random Hamiltonians in the last three decades. Technically, they are based on different mechanisms, and even a brief presentation of these approaches would require an entire book. Interested readers may familiarize themselves with the basic techniques in the monographs [9, 10]. Recently, there has been a wake of growing interest to the technique going back to the celebrated KAM (Kolmogorov‐Arnold‐Moser) theory originally developed for the analysis of stability of invariant tori in some nonlinear dynamical systems
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Recently, Imbrie [18] adapted the KAM techniques to the spectral analysis of random lattice Hamiltonians, in any dimension, and to one‐dimensional random spin chains.
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In essence, the “linear” version of the KAM method is an inductive, iterated use of the first‐order perturbation theory with an accurate account of the higher‐order terms represented by an infinite number of diagrams. At each step of the induction, one obtains an orthogonal basis for the considered (random) operator \n\n\nH\n\n(\nω\n)\n\n\n\n that is an approximate eigenbasis for the latter, but with better and better accuracy. The error terms on the \n\nk\n\n‐th step of induction feature a typical Newtonian decay rate like \n\n\n\ne\n\n−\n\nq\nk\n\n\n\n\n\n, where \n\n\nq\n∈\n\n(\n\n1\n,\n2\n\n)\n\n\n\n, which is not surprising since KAM technique is based on one or another form of Newton’s method. Once a new, more accurate eigenbasis of order \n\nk\n\n is constructed by perturbing its predecessor of order \n\n\n\n(\n\nk\n−\n1\n\n)\n\n\n\n, the matric elements of \n\n\nH\n\n(\nω\n)\n\n\n\n are computed in this (orthonormal) basis, and the process is repeated. KAM type constructions are usually quite complex and cumbersome. One has first to describe in detail the entire set of properties to be assumed on the step \n\n\nk\n−\n1\n≥\n0\n\n\n and then reproduced on the next induction step \n\nk\n\n.
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A synthetic method, employing some ideas of the KAM method and other, simpler techniques elaborated in the spectral theory of single‐particle random Hamiltonians, have been proposed first to the noninteracting random systems [11], and later on to their \n\nN\n\n‐particle counterparts. The pivot of this method, like in the KAM approach, is an accurate quantitative control of the “small denominators”—minimal distances between distinct random eigenvalues of random Hamiltonians associated with finite but growing subgraphs of a countable graph. It would be difficult to carry out such a program in the representation of distinguishable particles, i.e., on the Cartesian power \n\n\n\nG\nN\n\n\n\n.
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Now return to the semiquantitative analysis of a two‐particle Hamiltonian. Restrict ourselves to a case where the size of the underlying graph \n\nG\n\n, modeling the “physical” space where the two quantum particles evolve, has a fixed size, and allow us to vary the parameter \n\nϵ\n\n in \n\n\nϵ\n\n\nH\n\n0\n\n\n\n (the mobility amplitude), and to take it as small as needed for an attempt to make one step of application of the perturbation theory for nondegenerate spectra.
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With both \n\nϵ\n\n and \n\nh\n\n small enough, the main contribution comes from the random potential, so we have the eigenfunctions of the unperturbed operator \n\n\n\n\nΦ\n\n\nx\n,\ny\n\n\n\n\nΦ\n\n\nx\n,\ny\n\n\n=\n\n\n1\n\n\n\n{\nx\n}\n\n\n\n⊗\n\n\n1\n\n\n\n{\ny\n}\n\n\n\n\n\n with eigenvalues \n\n\n\nλ\n\nx\n,\ny\n\n\n=\n\nV\n\n\n(\n\nx\n,\nω\n\n)\n\n+\n\nV\n\n\n(\n\ny\n,\nω\n\n)\n\n\n\n, and we have to assess the difference between two such eigenvalues, labeled by two pairs of sites \n\n\n\n(\n\nx\n,\ny\n\n)\n\n,\n\n(\n\nx\n\'\n,\ny\n\'\n\n)\n\n\n\n of the graph \n\nG\n\n:
Since the potential is random, there can be no uniform, deterministic lower bound on the absolute value of the above difference: with positive probability, it can be smaller than any \n\n\nδ\n>\n0\n\n\n. The randomness, however, is a double‐edged sword: while small values of the difference are certainly possible, they may, or might, be unlikely, so we have to determine, how unlikely is to have \n\n\n|\n\nλ\n\nx\n,\ny\n\n\n−\n\nλ\n\n\nx\n\'\n\n,\n\ny\n\'\n\n\n\n|\n<\nδ\n\n\n.
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To begin with, notice that we have to deal with different eigenfunctions, hence with two nonidentical pairs \n\n\n\n(\n\nx\n,\ny\n\n)\n\n,\n\n(\n\nx\n\'\n,\ny\n\'\n\n)\n\n\n\n. Thus, \n\n\ncard\n\n{\n\nx\n,\ny\n\n}\n\n\n\n\n∩\n\n\n\n\n{\n\nx\n,\ny\n\n}\n\n≤\n1\n\n\n. Consider two cases.
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I. \n\n\ncard\n\n{\n\nx\n,\ny\n\n}\n\n\n\n\n∩\n\n\n\n\n{\n\nx\n\'\n,\ny\n\'\n\n}\n\n=\n0\n\n\n.
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In this case, the random variables \n\n\n\nλ\n\nx\n,\ny\n\n\n=\nV\n\n(\n\nx\n,\nω\n\n)\n\n+\nV\n\n(\n\ny\n,\nω\n\n)\n\n\n\n and \n\n\n\nλ\n\nx\n\'\n,\ny\n\'\n\n\n=\nV\n\n(\n\nx\n\'\n,\nω\n\n)\n\n+\nV\n\n(\n\ny\n\'\n,\nω\n\n)\n\n\n\n have no common terms, and therefore they are independent. Moreover, inside each pair we have independence, so the probability distribution of each sum can be easily obtained by convolution. For simplicity, consider the case of standard Gaussian variables, then each eigenvalue is also Gaussian with zero mean and variance \n\n2\n\n. The difference \n\n\n\nλ\n\nx\n,\ny\n\n\n−\n\nλ\n\nx\n\'\n,\ny\n\'\n\n\n\n\n is again a sum of two i.i.d. Gaussian variables \n\n\n\nλ\n\nx\n,\ny\n\n\n\n\n and \n\n\n−\n(\n\nλ\n\n\nx\n′\n\n,\n\ny\n′\n\n\n\n)\n\n\n, hence it is Gaussian with zero mean and variance \n\n4\n\n. Recalling the explicit form of the Gaussian probability density with variance \n\n\n\nσ\n2\n\n\n\n, which is uniformly bounded by \n\n\n1\n/\n\n\n2\nπ\n\nσ\n2\n\n\n\n\n\n, we conclude that
II. \n\n\ncard\n\n{\n\nx\n,\ny\n\n}\n\n\n\n\n∩\n\n\n\n\n{\n\nx\n\'\n,\ny\n\'\n\n}\n\n=\n1\n\n\n.
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Now the unordered pairs \n\n\nx\n,\ny\n\n\n and \n\n\n\nx\n\'\n\n,\ny\n\'\n\n\n have exactly one common point; let it be \n\nx\n\n, so we have a pair of eigenvalues \n\n\n\nλ\n\nx\n,\ny\n\n\n\n\n and \n\n\n\nλ\n\nx\n,\ny\n\'\n\n\n\n\n with \n\n\ny\n≠\ny\n\'\n\n\n and the difference given by
so it is again a sum of two independent random variables. Assuming as before the distribution to be Gaussian with zero mean and unit variance, we see that \n\n\n\nλ\n\nx\n,\ny\n\n\n−\n\nλ\n\n\nx\n\'\n\n,\n\ny\n\'\n\n\n\n\n\n is centered Gaussian with variance \n\n2\n\n, so the analog of Eq. (32) is now
The final conclusion is that for small \n\n\nδ\n>\n0\n\n\n, the small differences between any two eigenvalues corresponding to two distinct eigenfunctions on the symmetric square of the underlying graph is small, viz., of order of \n\n\nO\n\n(\nδ\n)\n\n\n\n. Therefore, for a finite graph of size \n\n\n\n|\nG\n|\n\n=\n\nn\n\n\n\n, the probability to have at least one pair of eigenvalues at distance smaller than \n\nδ\n\n is bounded by \n\n\n\n\n\nn\n\n\n(\n\n\nn\n\n−\n1\n\n)\n\nδ\n\n\n4\n\nπ\n\n\n\n=\nO\n\n(\n\n\n\nn\n\n2\n\nδ\n\n)\n\n\n\n (we used the weakest of the two estimates in Eqs. (32) and (33)).
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This simple probabilistic analysis provides the logical basis for the KAM approach, where we can rule out “small denominators” which cannot be tolerated in the analytic application of the first‐order perturbation formulae, at some initial scale. The rest of the procedure requires a number of analytic efforts, but the crucial point, viz., the possibility to avoid degenerate eigenvalues at the initial scale, is the direct consequence of the graph‐theoretical construction of a symmetric power of a graph \n\nG\n\n. Using a Cartesian power of \n\nG\n\n would at best significantly complicate the entire procedure, and perhaps render it impractical. In any case, no replacement for resorting to symmetric powers has been found so far in multiparticle localization theory of fermionic systems on graphs.
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4. Combinatorial and metric (quantum) graph
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Our final topic also concerns the constructive relations and interactions between the graph theory, in a broad sense, and the mathematical physics of the quantum world. However, the general direction of these interactions will be reversed, for we are going to discuss a very recently developed class of mathematical objects naturally emerged in the analysis of interacting quantum systems. We would like to attract the attention of experts in graph theory and related fields to the new area, where a number of questions are not even properly formulated, and many interesting phenomena are yet to be discovered.
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First, recall the notion of a metric graph; due to a wake of interest to the so‐called nanotubes, one often refers to these mathematical objects as “quantum graphs.”
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Metric graphs represent an important link between the discrete spaces and manifolds endowed with a rich local structure of a Euclidean space. By definition, a metric graph \n\n\nQ\n=\n\nQ\nΓ\n\n\n\n over a finite or countable unoriented combinatorial graph \n\n\n\n(\n\nΓ\n,\nE\n\n)\n\n,\n\n\n with the vertex set \n\nΓ\n\n and the edge set \n\nE\n\n, is a singular one‐dimensional manifold constructed as follows. Associate with each edge \n\n\ne\n=\n\n(\n\nι\n,\nτ\n\n)\n\n∈\nE\n\n\n an open interval \n\n\n\nI\ne\n\n\n\n, considered as a Riemannian manifold with the Riemannian metric inherited from \n\nR\n\n. In some models, all the intervals have the same lengths, so by a change of parameters one usually can assume they are replicas of \n\n\n\n(\n\n0\n,\n1\n\n)\n\n\n\n. In other models, on the contrary, one allows variable length of these basic intervals. We will assume the former and work with unit intervals. There is a canonical oriented graph associated with the unoriented graph \n\n\n\n(\n\nΓ\n,\nE\n\n)\n\n\n\n, with the same vertex set and two opposite edges for each edge in \n\nE\n\n. In some auxiliary constructions, this morphism from the category of unoriented graphs to that of oriented ones can be used, to avoid some ambiguities, but it will be not crucial to our purposes, since we will work with a second‐order differential operator (essentially the second derivative operator), so the orientation will not be really important.
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Each open interval \n\n\n\nI\ne\n\n≅\n\n(\n\n0\n,\n1\n\n)\n\n\n\n is canonically compactified by its natural embedding into \n\n\n\n[\n\n0\n,\n1\n\n]\n\n\n\n. Taking an edge \n\n\n\n(\n\nx\n,\ny\n\n)\n\n\n\n and fixing its orientation in one of the two possible ways, so that \n\n\n\n(\n\nx\n,\ny\n\n)\n\n≅\n\n(\n\nι\n,\nτ\n\n)\n\n\n\n, we thus can identify its starting point \n\nι\n\n with \n\n\n0\n∈\n\n[\n\n0\n,\n1\n\n]\n\n\n\n and the terminal point \n\nτ\n\n with \n\n\n1\n∈\n\n[\n\n0\n,\n1\n\n]\n\n\n\n. Next, we define the differential operator \n\n\nL\n=\n−\n\nd\n2\n\n/\nd\n\nt\n2\n\n\n\n in the space of twice differentiable functions on \n\n\n\n(\n\n0\n,\n1\n\n)\n\n\n\n; boundary conditions are discussed below. In other words, \n\n\nL\n=\n−\nΔ\n\n\n, where \n\nΔ\n\n is the Laplacian on the Riemannian manifold \n\n\n\n(\n\n0\n,\n1\n\n)\n\n\n\n. While \n\n\nd\n/\nd\nt\n\n\n requires a local coordinate, hence a fixed orientation, \n\nL\n\n is not sensible to this choice.
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Further, consider the disjoint union \n\n\n\nQ\nΓ\n\n\n(\n0\n)\n\n\n\n\n\n of the basic (open) intervals \n\n\n\nI\ne\n\n\n\n, finite or countable, with the natural structure of the measure space induced by the Lebesgue measure on each interval with the respective sigma‐algebra of measurable subsets. In turn, this allows us to introduce the Hilbert space of square‐integrable functions on \n\n\n\nQ\nΓ\n\n\n(\n0\n)\n\n\n\n\n\n; this is not yet an object we had needed, for there is no connections between the restrictions of a given function \n\nf\n\n on \n\n\n\nQ\nΓ\n\n\n(\n0\n)\n\n\n\n\n\n to different, pairwise disjoint connected components thereof.
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Now it is time to choose boundary conditions, having in mind the canonical embedding of \n\n\n\nQ\nΓ\n\n\n(\n0\n)\n\n\n\n\n\n into the union \n\n\n\nQ\nΓ\n\n\n\n of the compactified intervals \n\n\n\n\nI\n¯\n\ne\n\n≅\n\n[\n\n0\n,\n1\n\n]\n\n\n\n. In application to the “quantum” graphs, traditionally one imposes the Kirchhoff conditions. Now, for formal reasons, fix some orientation on each edge, hence, a local coordinate on each \n\n\n\n\nI\n¯\n\ne\n\n≅\n\n[\n\n0\n,\n1\n\n]\n\n\n\n. Then we can define the one‐sided first derivatives on each vertex, in the directions of all attached intervals \n\n\n\n\nI\n¯\n\ne\n\n\n\n. Let \n\n\n\nD\ne\n\n\n\n be such a derivative along the local coordinate on \n\n\n\n\nI\n¯\n\ne\n\n\n\n, and set \n\n\n\nc\ne\n\n=\n1\n\n\n for outgoing edges and \n\n\n\nc\ne\n\n=\n−\n1\n\n\n for the ingoing ones. The Kirchhoff conditions are as follows: a function \n\nf\n\n must be continuous at each vertex and obey a conservation law
Below we call the intervals \n\n\n\n\nI\n¯\n\ne\n\n\n\ncontinuous edges.
\n
Now, using the standard methods of functional analysis, one can construct a self‐adjoint extension of the “Laplacian” \n\nL\n\n with Kirchhoff boundary conditions, and for any, say, bounded measurable function \n\n\n\nV\n\n:\n\nQ\nΓ\n\n→\nR\n\n\n (a potential), define the Schrödinger operator \n\n\n\n\nH\n\n\nV\n\n\n\n\n as unbounded self‐adjoint operator in \n\n\n\nH\n\n=\n\nL\n2\n\n\n(\n\n\nQ\nΓ\n\n\n)\n\n\n\n, with the suitable domain.
\n
One can perturb the above, rather idyllic picture in several ways. First, one can consider a random potential \n\n\n\nV\n\n\n(\nω\n)\n\n\n\n, taking i.i.d. random values on each edge. Further, on can vary the lengths \n\n\n\nl\ne\n\n\n\n of the continuous edges, assuming that \n\n\n\nl\ne\n\n\n(\nω\n)\n\n\n\n are i.i.d. random variables with a common probability distribution. From the functional analytic point of view, treating unbounded self‐adjoint operators on metric graphs, in the framework of random operators, is substantially more delicate a matter that the analysis of finite‐difference operators on the underlying discrete, combinatorial graphs. One may wonder, whether some properties of the Hamiltonians on the underlying graph can be useful for the analysis of their continuous siblings \n\n\n\nQ\nΓ\n\n\n\n. The theory of boundary triples (cf. [19]) provides a powerful and valuable tool of spectral analysis on continuous metric graphs, where an essential part of technical work is carried out in a simpler framework of countable graphs with discrete Schrödinger operators.
\n
Now we turn to a further development in this direction made recently by Sabri [7] who introduced the notion of multiparticle quantum graph. We consider the simplest nontrivial case of \n\n\nN\n=\n2\n\n\n quantum particles on a quantum graph \n\n\n\nQ\nΓ\n\n\n\n. To be able to refer to existing results and publications, we assume the particles distinguishable.
\n
In the discussion of two‐particle systems on a graph \n\nG\n\n in Section 3, the pair \n\n\n\n(\n\n\nx\n1\n\n,\n\nx\n2\n\n\n)\n\n\n\n was ranging in the Cartesian (and then symmetric) square of \n\nG\n\n, and the latter is, topologically, a discrete space, thus essentially of the same nature as the factors in the product \n\n\nG\n×\nG\n\n\n. But now that the configuration space \n\n\n\nQ\nΓ\n\n\n\n for each particle is a continuous object, viz. a (singular) one‐dimensional Riemannian manifold, the situation changes radically: the configuration space for the pair \n\n\n\n(\n\n\nx\n1\n\n,\n\nx\n2\n\n\n)\n\n\n\n is locally a two‐dimensional manifold; in the case of an \n\nN\n\n‐tuple \n\n\n\n(\n\n\nx\n1\n\n,\n…\n,\n\nx\nN\n\n\n)\n\n\n\n it becomes \n\nN\n\n‐dimensional. Many specifically 1D methods of spectral analysis are inapplicable in dimension \n\n\nd\n≥\n2\n\n\n.
\n
Shortly after the publication of the first results on \n\nN\n\n‐particle Anderson localization in periodic lattices and in Euclidean spaces, Sabri [7] proposed an interesting extension of the new techniques and results to the multiparticle systems on quantum graphs. His construction was essentially motivated by a specific goal, but there are various contexts where the construction itself may prove valuable.
\n
For \n\n\nN\n=\n2\n\n\n, one has to start again with the building blocks of a 1D quantum graph \n\n\n\nQ\nΓ\n\n\n\n over a combinatorial graph \n\nΓ\n\n: the open finite intervals associated with each edge of \n\nΓ\n\n. Restricting the positions \n\n\n\nx\ni\n\n\n\n of the two particles to their respective continuous edges identified with \n\n\n\n[\n\n0\n,\n1\n\n]\n\n\n\n, we have the pair \n\n\n\n(\n\n\nx\n1\n\n,\n\nx\n2\n\n\n)\n\n\n\n ranging in the unit square \n\n\n\n[\n\n0\n,\n1\n\n]\n\n×\n\n[\n\n0\n,\n1\n\n]\n\n\n\n. For brevity, call such basic squares cells. Each cell is delimited by four continuous edges inherited from the first and from the second particle, and these four (continuous) edges are the loci of contact between the cells. Clearly, this complicates the structure of what was the edge set in the underlying graph \n\nΓ\n\n, but this is the natural replacement for the notion of the edge set; this is what defines the topology and metric geometry of the new object, called by Sabri a \n\n\ntwo\n\n\n‐particle (more generally, \n\nN\n\n‐particle) quantum graph \n\n\n\nQ\nΓ\n\n\n(\n2\n)\n\n\n\n\n\n.
\n
Just like the Laplacian \n\nL\n\n defined on the quantum graph, we can define its two‐particle counterpart \n\n\n\nL\n\n\n(\n2\n)\n\n\n\n\n\n : first, on the unit squares, and then proceed to self‐adjoint extensions with one or another kind of boundary conditions to be imposed on the 1D continuous edges of the conventional, \n\n\none\n\n\n‐particle quantum graphs supporting each of the two particles. This inevitable functional analytic work has been done by Sabri. And of course, once the natural Laplacian \n\n\n\nL\n\n\n(\n2\n)\n\n\n\n\n\n is defined as an unbounded self‐adjoint operator with a suitable domain in the Hilbert space of square‐integrable functions on \n\n\n\nQ\nΓ\n\n\n(\n2\n)\n\n\n\n\n\n, one can also define the Schrödinger operators \n\n\n\n\nH\n\n\nV\n\n\n\n(\n2\n)\n\n\n\n=\n−\n\nL\n\n\n(\n2\n)\n\n\n\n+\nV\n\n\n, e.g., for bounded measurable functions \n\nV\n\n. In Figure 2, we give an example of three models based on the same graph structure: a combinatorial graph, a quantum graph, and a two‐dimensional domain surrounding the quantum graph in question.
\n
Figure 2.
Example of (a) physical, thin two‐dimensional area \n\nA\n\n; (b) corresponding metric graph \n\n\n\nQ\nΓ\n\n\n\n: a mathematical abstraction where the finite width of \n\nA\n\n is ignored; and (c) the combinatorial graph with the same vertices as \n\n\n\nQ\nΓ\n\n\n\n.
\n
The new construction raises a number of questions, of different nature. One of them concerns the constructive relations between the spectral properties of a Schrödinger operator \n\n\n\n\nH\n\n\nV\n\n\n\n(\n2\n)\n\n\n\n\n\n on the continuous, locally two‐dimensional (2D) manifold \n\n\n\nQ\nΓ\n\n\n(\n2\n)\n\n\n\n\n\n, and its analog on the Cartesian square of the combinatorial graph \n\nΓ\n\n.
\n
Another question, of functional analytic nature, raised by Sabri, concerns an explicit description of the self‐adjoint extensions of the 2D Laplacian initially defined, say, on infinitely differentiable functions with support inside an open cell \n\n\n≅\n\n\n\n(\n\n0\n,\n1\n\n)\n\n\n2\n\n\n\n. It appears that the corner points make the explicit analysis difficult, although the existence of the desired extensions poses no serious problem.
\n
\n
\n
5. Conclusion
\n
Mathematical modeling of physical phenomena had provided important motivations for developing various fields of mathematical physics since several centuries; as to the quantum physics, its development was from the beginning of the twentieth century parallel to the development of the functional analysis in general and spectral theory of operators in particular. The remarkable discovery made by P. W. Anderson in 1958 brought to life a synthetic approach to modeling disordered systems based on a fusion of analysis in a broad sense with probability theory. The physical community came to realize that the models based on the idealized picture of perfectly periodic crystals miss some crucial mechanisms responsible for transport (e.g., electrical conductivity) or absence thereof under the Anderson localization. The classical formulae for conductivity and many related phenomena, crucial for the development of modern microelectronics and nanotechnologies, cannot ignore the localization/delocalization problematics. While the most simple models may refer to the integer (and some other periodic) lattices in a Euclidean space where classical Fourier analysis can use the method of separation of variables, the situation can be significantly more complex in the case of quasicrystals, featuring both a local order and long‐range disorder. Mathematically, such structures are described as nonperiodic graphs where the Fourier analysis breaks down, and one needs some efficient, constructive replacements. Furthermore, large and complex molecules studied in organic chemistry and molecular biology also require a versatile toolbox not limited to a commutative Fourier analysis. Also, the crystalline media in presence of structural (e.g., mechanical) defects are not stricto sensu periodic, so again one needs robust eigenvalue distribution bounds not relying on the exactly periodic geometry of the media. In Section 2, we have seen that the isoperimetric inequalities, appeared in the graph theory under the influence of its diverse applications, provide indeed adequate tools for an asymptotic analysis of the limiting eigenvalue distribution for discrete quantum Hamiltonians used in physics in the framework of the so‐called tight‐binding approximation effective for the “low” energies. The term “low” actually refers to the energies lost important to the quantum processes exploited in modern microdevices (e.g., CPU having diameter of a few millimeters and width of order of a few dozens of atomic layers), in biological tissues and technologically created organic substances.
\n
In 2008, The Isaac Newton Institute for Mathematical Sciences in Cambridge, Great Britain, has organized a semiannual program “Mathematics and Physics of Anderson Localization: 50 Years Later” aiming to summarize the impact of Anderson’s theory on physical theories and applications as well as on the mathematical physics. The general conclusion many participants and younger researchers could draw from numerous and diverse presentations was that the paradigm of quantum localization/delocalization provides today both a language and a general theoretic background for many specific directions of research; it is not an isolated pragmatic physical model or abstract mathematical problem. The program in question also revealed to the physics and mathematics communities the importance of the interparticle interaction which was briefly discussed in Section 3; the need for such theory was emphasized already by Anderson in his early papers, but one had to wait almost half a century to see its emergence in independent physical and rigorous mathematical works. Shortly after the program, the Anderson localization theory for interactive disordered systems has been applied (in mathematical works) to the nanotubes modeled by quantum graphs. While the size limitations of the present work do not allows us to present mathematical details of the new theory, there is no doubt that many of its mathematical aspects are closely related to the methods of the graph theory. Further reading, along with an extensive bibliography, can be found in the first monograph [4] dedicated to localization phenomena in interactive systems. This new direction of mathematical physics still is at its early stage of development. The language and toolbox of the graph theory proved to be very useful here, as we have seen in Section 3. On the other hand, new structures discussed in Section 4, emerging from the analysis of multiparticle quantum graphs open new problems and propose new types of models to the graph theory. This chapter was written in the hope to bring closer the communities of researchers, particularly the younger ones, working in functional analysis, graph theory in a broad sense, and in probability theory.
\n
\n\n',keywords:"isoperimetric estimates, Cheeger bound, Lifshitz tails, Anderson localization, multiparticle localization, quantum graphs",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/55199.pdf",chapterXML:"https://mts.intechopen.com/source/xml/55199.xml",downloadPdfUrl:"/chapter/pdf-download/55199",previewPdfUrl:"/chapter/pdf-preview/55199",totalDownloads:834,totalViews:219,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"October 18th 2016",dateReviewed:"March 13th 2017",datePrePublished:"December 20th 2017",datePublished:"January 31st 2018",dateFinished:"May 7th 2017",readingETA:"0",abstract:"This chapter is devoted to various interactions between the graph theory and mathematical physics of disordered media, studying spectral properties of random quantum Hamiltonians. We show how the notions, methods, and constructions of graph theory can help one to solve difficult problems, and also highlight recent developments in spectral theory of multiparticle random Hamiltonians which both benefit from graph‐theoretical methods and suggest original structures where new insights are required from various areas of mathematical physics in a broad sense.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/55199",risUrl:"/chapter/ris/55199",book:{slug:"graph-theory-advanced-algorithms-and-applications"},signatures:"Victor Chulaevsky",authors:[{id:"198567",title:"Prof.",name:"Victor",middleName:null,surname:"Chulaevsky",fullName:"Victor Chulaevsky",slug:"victor-chulaevsky",email:"victor.tchoulaevski@univ-reims.fr",position:null,institution:{name:"University of Reims Champagne-Ardenne",institutionURL:null,country:{name:"France"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Isoperimetric bounds, spectral gaps, and quantum localization",level:"1"},{id:"sec_3",title:"3. Symmetric powers of graphs and spectra of fermionic systems",level:"1"},{id:"sec_3_2",title:"3.1. Motivation and preliminaries",level:"2"},{id:"sec_4_2",title:"3.2. Construction of a symmetric power of a graph",level:"2"},{id:"sec_4_3",title:"3.2.1. An example in one dimension",level:"3"},{id:"sec_5_3",title:"3.2.2. General construction",level:"3"},{id:"sec_6_3",title:"3.2.3. Hilbert space isomorphism: antisymmetric functions versus symmetric power",level:"3"},{id:"sec_7_3",title:"3.2.4. An application: KAM‐type analysis of two‐particle fermionic random Hamiltonians",level:"3"},{id:"sec_10",title:"4. Combinatorial and metric (quantum) graph",level:"1"},{id:"sec_11",title:"5. Conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'Anderson PW. Absence of diffusion in certain random lattices. Physical Review. 1958;109(5):1492‐1505\n'},{id:"B2",body:'Basko DM, Aleiner IL, Altshuler BL. Metal‐insulator transition in a weakly interacting many‐electron system with localized single‐particle states. Annalen der Physik. 2006;321(5):1126‐1205\n'},{id:"B3",body:'Chulaevsky V, Suhov Y. Eigenfunctions in a two‐particle Anderson tight binding model. Communications in Mathematical Physics. 2009;289:701‐723\n'},{id:"B4",body:'Chulaevsky V, Suhov Y. Multi‐particle Anderson localisation: Induction on the number of particles. Mathematical Physics Analysis and Geometry. 2009;12:117‐139\n'},{id:"B5",body:'Aizenman M, Warzel S. Localization bounds for multiparticle systems. Communications in Mathematical Physics. 2009;290:903‐934\n'},{id:"B6",body:'Chulaevsky V, Suhov Y. Multi‐scale Analysis for Random Quantum Systems with Interaction. Boston: Birkhäuser; 2013. p. 236\n'},{id:"B7",body:'Sabri M. Anderson localization for a multi‐particle quantum graph. Reviews in Mathematical Physics. 2014;26(1). DOI: 10.1142/S0129055X13500207.\n'},{id:"B8",body:'Lifshitz IM, Gredescul SA, Pastur LA. Introduction to the Theory of Disordered Systems. New York: Wiley; 1988. p. 358\n'},{id:"B9",body:'Carmona R, Lacroix J. Spectral Theory of Random Schrödinger Operators. Boston, Basel, Berlin: Birkhäuser; 1990. p. 587\n'},{id:"B10",body:'Stollmann P. Caught by Disorder. Boston: Birkhäuser; 2001. p. 166\n'},{id:"B11",body:'Chulaevsky V. Direct scaling analysis of localization in single‐particle quantum systems on graphs with diagonal disorder. Mathematical Physics Analysis and Geometry. 2012;15:361‐399\n'},{id:"B12",body:'Temple G. The theory of Rayleigh’s principle as applied to continuous systems. Proceedings of the Royal Society of London Series A. 1928;19:276‐293\n'},{id:"B13",body:'Chung FRK. Spectral Graph Theory. Washington, DC: CBMS Conference Series in Mathematics; 1997. p. 117\n'},{id:"B14",body:'Chung FRK, Grigor’yan A, Tau S‐T. Upper bounds for eigenvalues of discrete and continuous Laplace operators. Advances in Mathematics. 1996;117(2):165‐178\n'},{id:"B15",body:'Wegner F. Bounds on the density of states in disordered systems. Zeitschrift Fur Physik B Condensed Matter. 1981;44:9‐15\n'},{id:"B16",body:'Chulaevsky V. From fixed‐energy localization analysis to dynamical localization: An elementary path. Journal of Statistical Physics. 2014;154:1391‐1429\n'},{id:"B17",body:'Combes J‐M, Thomas L. Asymptotic behaviour of eigenfunctions for multi‐particle Schrödinger. Communications in Mathematical Physics. 1973;34:251‐263\n'},{id:"B18",body:'Imbrie J. Multi‐scale Jacobi method for Anderson localization. Communications in Mathematical Physics. 2016;341:491‐521\n'},{id:"B19",body:'Brüning J, Geyler V, Pankrashkin K. Spectra of self‐adjoint extensions and applications to solvable Schrödinger operators. Reviews in Mathematical Physics. 2008;20(20):1‐70\n'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Victor Chulaevsky",address:"victor.tchoulaevski@univ‐reims.fr",affiliation:'
Department of Mathematics, University of Reims, Reims Cedex, France
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Frasch",authors:[{id:"14757",title:"Prof.",name:"Wayne",middleName:null,surname:"Frasch",fullName:"Wayne Frasch",slug:"wayne-frasch"},{id:"317054",title:"Prof.",name:"Michael",middleName:null,surname:"Kuby",fullName:"Michael Kuby",slug:"michael-kuby"},{id:"317055",title:"Dr.",name:"Fusheng",middleName:null,surname:"Xiong",fullName:"Fusheng Xiong",slug:"fusheng-xiong"}]},{id:"72140",title:"Comparative Study of Algorithms Metaheuristics Based Applied to the Solution of the Capacitated Vehicle Routing Problem",slug:"comparative-study-of-algorithms-metaheuristics-based-applied-to-the-solution-of-the-capacitated-vehi",signatures:"Fernando Francisco Sandoya Sánchez, Carmen Andrea Letamendi Lazo and Fanny Yamel Sanabria Quiñónez",authors:[{id:"155426",title:"Ph.D.",name:"Fernando",middleName:"Francisco",surname:"Sandoya",fullName:"Fernando Sandoya",slug:"fernando-sandoya"},{id:"313162",title:"M.Sc.",name:"Carmen",middleName:null,surname:"Letamendi",fullName:"Carmen Letamendi",slug:"carmen-letamendi"},{id:"319376",title:"Dr.",name:"Fanny",middleName:null,surname:"Sanabria",fullName:"Fanny Sanabria",slug:"fanny-sanabria"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"62502",title:"Thiophene S-Oxides",doi:"10.5772/intechopen.79080",slug:"thiophene-s-oxides",body:'\n
\n
1. Early history of oxidation reactions of thiophenes: cycloaddition reactions of thiophene S-oxides prepared in situ in absence of Lewis acids
\n
In the first half of the 20th century, considerable effort was devoted to the oxidation of the heteroaromatic thiophene (1) with the understanding that the oxidation of thiophene to thiophene S,S-dioxide (2) (Figure 1) would be accompanied by the loss of aromaticity [1, 2]. The non-substituted thiophene S,S-dioxide (1) is not very stable in the pure state [3], but undergoes a slow dimerization with concurrent extrusion of SO2 from the primary cycloadduct (4) [4], leading to 5 (Scheme 1). Only much later were the properties and reactivity of pure, isolated non-substituted thiophene S,S-dioxide (2) described [5].
\n
Figure 1.
Structure of thiophene (1) and oxygenated thiophenes 2 and 3.
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Scheme 1.
Dimerisation of unsubstituted thiophene S,S-dioxide (2).
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Much of the early work on the oxidation of thiophenes to thiophene S,S-dioxides involved hydrogen peroxide (H2O2) as oxidant, later meta-chloroperoxybenzoic acid (m-CPBA). That thiophene S-oxide was an intermediate in such oxidation reactions [6, 7, 8] was evident from the isolation of so-called sesquioxides as dimerization products of thiophene S-oxides [9, 10, 11, 12]. Here, the thiophene S-oxide acted as diene with either another molecule of thiophene S-oxide or thiophene S,S-dioxide acting as ene [9, 10, 11, 12] to give cycloadducts 6–8 (Figure 2). Thiophene S-monoxide (3) as an intermediate in the oxidation process of thiophene (1) to thiophene S,S-dioxide (2) could not be isolated under the conditions.
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Figure 2.
Sesquioxides obtained by dimerization of elusive thiophene S-oxide and by cycloaddition of thiophene S-oxide to thiophene S,S-dioxide.
\n
Nevertheless, the idea that a thiophene S-oxide intermediate could be reacted with an alkene of choice led Torssell [13] oxidize methylated thiophenes with m-CPBA in the presence of quinones such as p-benzoquinone (12). This gave cycloadducts 13 and 14 (Scheme 2) [13]. Further groups [11, 12, 14, 15, 16, 17, 18, 19] used this strategy to react thiophene S-oxides such as 11, prepared in-situ with alkenes and alkynes in [4 + 2]-cycloadditions (Schemes 3 and 4). In the reaction with alkenes, 7-thiabicyclo[2.2.1]heptene S-oxides such as 13 were obtained, while the reaction of thiophene S-oxides with alkynes led to cyclohexadienes and/or to aromatic products, where the initially formed, instable 7-thiabicyclo[2.2.1]hepta-2,5-diene S-oxide system 21 extrudes its SO bridge spontaneously (Scheme 4). A number of synthetic routes to multifunctionalized cyclophanes 32 [17], aryl amino acids 25 [16] and to crown ethers 29 [15] (Scheme 5) have used the cycloaddition of thiophene S-oxides 19, created in-situ, as a key step. The formation of the 7-thiabicyclo[2.2.1]heptene S-oxides (such as 13, 18) proceeds with stereocontrol. The cycloadditions yield predominantly endo-cycloadducts, with the oxygen of the sulfoxy bridge directed towards the incoming dienophile, exhibiting the syn-π-facial stereoselective nature of the reaction (see below for further discussion of the stereochemistry of the cycloadducts). Thiophene S,S-dioxides 2 possess an electron-withdrawing sulfone group, which leads both to a polarization and to a reduction of the electron density in the diene [20]. This results in a decrease of the energy of the HOMO as compared to identically substituted cyclopentadienes [20]. Thiophene S,S-dioxides 2 are sterically more exacting than C5 non-substituted cyclopentadienes, with the lone electron pairs on the sulfone oxygens leading to adverse non-bonding interactions with potentially in-coming dienophiles of high π-electron density. Thus, thiophene S,S-dioxides 2 often require higher temperatures [21, 22] in cycloaddition reactions than identically substituted cyclopentadienes. Recent frontier molecular orbital calculations at the HF/6-311++G(d,p)//M06-2X/6-31+G(d) level theory have shown that both HOMO (by 0.5 eV) and LUMO (by 0.4 eV) in thiophene S-oxide (3) are slightly higher in energy than in thiophene S,S-dioxide (2) [23].
\n
Scheme 2.
Thiophene S-oxide (11), created in situ, reacts in Diels-Alder type fashion with p-benzoquinone (12).
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Scheme 3.
Cycloaddition of thiophene S-oxides, prepared in situ, with alkenes.
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Scheme 4.
Cycloaddition of thiophene S-oxides (19), prepared in situ, with alkynes.
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Scheme 5.
Cycloaddition of thiophene S-oxides prepared in situ—applications in the synthesis of functionalized aminocarboxylic acids 25, crown ethers 29 and cyclophanes 32.
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Oxidation of the thienyl-unit in 33 leads to an intramolecular cycloaddition, where indanones 34 are obtained (Scheme 6) [24].
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Scheme 6.
Intramolecular cycloaddition of in situ prepared thiophene S-oxide 34.
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\n
\n
2. Cycloaddition reactions of thiophene S-oxide prepared in situ in the presence of Lewis acids: thiophene S-oxides are isolated
\n
Yields of cycloadducts have been found to be much higher, when oxidative cycloaddition reactions of thiophenes are carried out with meta-chloroperoxybenzoic acid (m-CPBA) or with H2O2 at lower temperatures such as at −20°C in the presence of a Lewis acid catalyst such as BF3·Et2O [11, 12, 25, 26] (Scheme 7) or of trifluoroacetic acid (CF3CO2H) [27]. Electron-poor dienophiles such as tetracyanoethylene, acetylene dicarboxylates, quinones, maleimides and maleic anhydride and mono-activated enes such as cyclopentenone and acrolein were used in these reactions.
\n
Scheme 7.
Oxidative cycloaddition of thiophene 36 to naphthoquinone (37) in the presence of BF3.Et2O.
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Under the conditions m-CPBA/BF3·Et2O, the cycloadditive transformation of thiophene S-oxides, prepared in situ, was used in the synthesis of new cyclophanes such as 39 (Scheme 8) [25]. A series of 2,3-bis(hydroxyphenyl) substituted 7-thiabicyclo[2.2.1]hept-2-ene S-oxides as potential estrogen receptor ligands were prepared by oxidative cycloaddition of 3,4-bis(hydroxyphenyl)thiophenes in the presence of BF3·Et2O [28]. Also the key step in Yu et al.’s [27] synthesis of steroidal saponins 44, closely related to the E-ring areno containing natural products aethiosides A–C, is a BF3·Et2O catalyzed oxidative cycloaddition of the thieno-containing steroidal saponin 42 (Scheme 9) [26]. Furthermore, Zeng and Eguchi [29] were able to functionalize C60 (46) by cycloaddition with in-situ produced 2,5-dimethylthiophene S-oxide (45) [29, 30] (Scheme 10). Nevertheless, sterically hindered thiophenes are more difficult to be subjected to the oxidative cycloaddition reactions (Figure 3).
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Scheme 8.
Preparation of multifunctionalized cyclophane 41 by oxidative cycloaddition of thiophenophane 39 in the presence of BF3.Et2O.
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Scheme 9.
Preparation of aethiosides A–C (44a–c) by oxidative cycloaddition of thienosteroidal sapogenin 42.
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Scheme 10.
Cycloaddition of 2,5-dimethylthiophene S-oxide (45), prepared in situ, to C60 (46).
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Figure 3.
Orthothiophenophanes 48 and 49 do not allow for enough reaction volume and do not undergo oxidative cycloadditions with either alkynes or alkenes under the conditions (m-CPBA, BF3.Et2O, CH2Cl2) [31].
\n
\n
\n
3. Preparation and isolation of pure thiophene S-oxides
\n
Thiophene S-oxides could be isolated in pure form as side-products in a number of oxidative cycloaddition reactions using alkylated thiophenes as substrates run with m-CPBA in the presence of BF3·Et2O [11, 12]. Nevertheless, the first ascertained thiophene S-oxide (51) isolated in pure form came from the oxidation of the sterically exacting 2,5-bis-tert-butylthiophene (50) in absence of a Lewis acid or an added protic acid. 2,5-Bis-tert-butylthiophene S-oxide (51) could be isolated in 5% yield [32] (Scheme 11).
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Scheme 11.
Isolation of 2,5-bis-tert-butylthiophene S-oxide 51 by simple thiophene oxidation with meta-chloroperoxybenzoic acid (m-CPBA) [32].
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Previous to the isolation of thiophene S-oxides in pure form, based on UV-spectroscopic measurements, Procházka [33] had claimed that the parent thiophene S-oxide (3) could be prepared by double elimination from 3,4-dimesyloxy-2,3,4,5-tetrahydrothiophene S-oxide (53) and studied in solution. While subsequently the latter part of the assertion was thrown into doubt, the isolation of sesquioxides 7/8 from the reaction indicated at least the presence of thiophene S-oxide under these conditions [33] (Scheme 12).
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Scheme 12.
In situ preparation of parent thiophene S-oxide (3) by an elimination reaction [33].
\n
Interestingly, a toluene solution of η5-ethyltetramethylcyclopentadienyl-η4-tetramethylthienyl rhodium complex [Cp*Rh(η4-TMT)] (54) can be oxidized with dry oxygen to [Cp*Rh(TMTO)] (56), which features a η4-coordinated thiophene S-oxide ligand. Complex 56 was isolated and an X-ray crystal structure was carried out. Alternatively, [Cp*Rh(η4-TMT)] (54) can be oxidized electrochemically to [Cp*Rh(η4-TMT)]2+ (55), which can also be obtained by protonation of [Cp*Rh(TMTO)] (56). Reaction of [Cp*Rh(η4-TMT)]2+ (55) with potassium methylsilanolate (KOSiMe3) leads back to [Cp*Rh(TMTO)] (56) [34] (Scheme 13).
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Scheme 13.
Oxidation of [Cp*Rh(η4-TMT)] (54) to [Cp*Rh(TMTO)] (56) [34].
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The reaction of the cationic transitory ruthenium complex [Ru(C6R6)(C4R4S)]+ (57) with hydroxyl anion (OH−) gives Ru(C6H6)(C4R4SO) (58) [35] (Scheme 14). Here, in contrast to the complex [Cp*Rh(TMTO)] (56), the thiophene S-oxide ligand in Ru(C6H6)(C4R4SO) (58) is not stable, but opens to an acetylpropenethiolate. Stable osmium thiophene S-oxide complexes of type (cymene)Os(C4Me4S=O) have also been prepared [36]. In neither of the cases, was it tried to decomplex the thiophene S-oxide ligand.
\n
Scheme 14.
Base hydrolysis of [Ru(C6R6)(C4R4S)]+ (57) [34].
\n
In the 1990s, two main synthetic methodologies were developed to prepare thiophene S-oxides 63. The first involves the reaction of substituted zirconacyclopentadienes 62 with thionyl chloride (SOCl2), developed by Fagan et al. [37, 38] and by Meier-Brocks and Weiss [39]. Typically, tetraarylzirconacyclopentadienes 62a can be synthesized easily by reacting CpZrCl2 (59), n-BuLi and diarylethyne (61a) in one step (Scheme 15). This strategy was followed by Tilley et al. [40, 41] in their synthesis of substituted thiophene S-oxides. Miller et al. published results for a synthesis of 2,5-diarylthiophene S-oxides (63b) along the same lines, using ethynylarene (61b) [42].
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Scheme 15.
Synthesis of tetraarylthiophene S-oxides 63a/b by reaction of tetraarylzirconacyclopentadienes 62a/b with SOCl2.
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The other methodology involves an oxidation of a thiophene with either a peracid in the presence of a Lewis acid such as titanium tetrachloride (TiCl4) [43] or boron trifluoride etherate (BF3·Et2O) [44, 45] or with hydrogen peroxide in the presence of a protonic acid such as trifluoroacetic acid [46, 47] (Scheme 16). Also, the use of the reaction system H2O2 in presence of NaFe(III) ethylenediaminetetraacetate/Al2O3 has been reported [48, 49] (Scheme 16) as has been the use of the reaction system [(C18H37)2(CH3)2N]3[SiO4H(WO5)3] [50]. The thiophene S-oxides 65, suitably substituted, can be isolated by column chromatography and can be held in substance for a number of weeks without appreciable degradation, when in crystallized form and when kept in the dark. It is supposed that the Lewis acid not only activates the peracid, but also coordinates to the oxygen in the formed thiophene S-oxide, thus reducing the electron-density on the sulfur of the thiophene S-oxide, making it less prone to undergo a second oxidation to the thiophene S,S-dioxide.
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Scheme 16.
Preparation of thiophene S-oxides 65 by oxidation of thiophenes 64 in the presence of a Lewis acid or a protonic acid.
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It has been shown that in a molecule, such as 66 or 67, with two thienyl cores, both can be oxidized to thienyl-S-oxides with m-CPBA, BF3·Et2O CH2Cl2, −20°C) [11, 17] . Under these conditions, the second thiophene unit can compete successfully with a thiophene S-oxide for the oxidant (Figure 4).
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Figure 4.
Known bisthienyl-S-oxides 66 and 67.
\n
\n
\n
4. Reactions of thiophene S-oxides
\n
\n
4.1. [4 + 2]-cycloaddition reactions
\n
Even before thiophene S-oxides could be isolated in pure form, it was evident that thiophene S-oxides are good dienes in cycloaddition reactions, as “trapping” by cycloaddition reaction was one of the standard techniques to gauge the presence of thiophene S-oxide intermediates and provided a versatile preparative entry to 7-thiabi-cyclo[2.2.1]heptene S-oxides 68. These in turn could be converted to substituted arenes 71 by either pyrolysis [15], photolysis [51], or PTC-catalyzed oxidative treatment with KMnO4 [15] or electrochemical oxidation [18] or 7-thiabicyclo-[2.2.1]heptenes (70) by reaction of 68 with PBr3 [52]. Reaction of 68 with tributyltin hydride gives cyclic dienes such as 72 [▬X▬X▬ = ▬(CO)N▬Ph(CO)▬]. Base catalyzed cleavage of the sulfoxy bridge of 1,4-dihalo-7-thiabicyclo[2.2.1]heptane S-oxides 68 (R1 = Cl or Br) leads to the generation of diaryl disulfides such as 69 (Scheme 17).
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Scheme 17.
7-Thiabicyclo[2.2.1]heptene S-oxides 68 as versatile precursors to arenes.
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With the possibility of isolating the thiophene S-oxides, it became possible to carry out cycloaddition reactions with alkenes that themselves react with m-CPBA. Thiophene S-oxides such as 73 have been found to react equally well with electron-rich alkenes such as enol ethers (74) [53], with electron neutral alkenes such as with cyclopentene (76) [53, 54] and with electron-poor alkenes such as with cyclopentenone or with maleic anhydride [11, 54] (Scheme 18). Also, thiophene S-oxides react with bicyclopropylidene (82) [55] under high pressure (10 kBar, Scheme 19), with allenes [56] (such as 79, Scheme 19), with cyclopropylideneketone [55] (Scheme 20) and with benzyne (90) [56], both formed in-situ (Scheme 21). The reaction of tetrachlorocyclopropene (93) with 3,4-bis-tert-butylthiophene S-oxide (73) led to 6,7-bis-tert-butyl-2,3,4,4-tetrachloro-8-thiabicyclo[3.2.1]octa-2,6-diene 8-oxide (95), resulting from a ring opening of the primary cycloadduct 94 with a concomitant migration of a chloro atom [57] (Scheme 22). The ability of the thiophene S-oxides to undergo cycloadditions with alkenes, regardless of the electron demand of the reaction, has made Houk et al. say that thiophene 1-oxide cycloadditions warrant their classification as click reactions [23].
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Scheme 18.
3,4-Bis-tert-butylthiophene S-oxide (73) cycloadding to electron-rich and electron-neutral alkenes.
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Scheme 19.
Thiophene S-oxides cycloadd to allenes and to bicyclopropylidene (82) under high pressure.
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Scheme 20.
One pot Wittig reaction—Diels Alder reaction with thiophene S-oxide 87 as diene.
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Scheme 21.
Cycloaddition of thiophene S-oxide 91 with benzyne (90), prepared in situ.
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Scheme 22.
Cycloaddition of thiophene S-oxide (73) with tetrachlorocyclopropene (93).
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Thiophene S-oxides are good precursors for the preparation of heavily substituted arenes such as 100 [58] (Scheme 23). Often, tetraarylcyclopentadienones 97 are used to synthesize oligoaryl benzenes by cycloaddition reaction. However, tetraphenylthiophene S-oxide (96) is the more reactive diene when compared to tetraphenylcyclopentadienone (97) as can be seen in the competitive cycloaddition of 96 and 97 with N-phenylmaleimide (98), where at room temperature only tetraphenylthiophene S-oxide undergoes cycloaddition to give 99 (Scheme 23) [58]. 99 can be converted to the heavily substituted phthalimide 100 [58], either by extruding the SO group thermally in diphenyl ether (Scheme 23) or by reaction with KMnO4/PTC.
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Scheme 23.
Thiophene S-oxide 96 competes efficiently with tetracyclone 97 for N-phenylmaleimide (98).
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Sometimes, tetraphenylthiophene S-oxide (96) and tetraphenylcyclopentadienone (97) give different products in cycloaddition reactions. A typical example is their cycloaddition to benzo[b]thiophene S,S-dioxide (101), where the reaction with 96 leads to the formation of dibenzothiophene S,S-dioxide 102, but with 97 gives dibenzothiophene 104 [59] (Scheme 24). The reason for this difference lies in the tendency of tetracyclines such as 94 to be oxidized to pyrones 102 at higher reaction temperatures, with the S,S-dioxides playing the oxidizing agent [59] (Scheme 24).
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Scheme 24.
Comparison of the cycloaddition of tetraphenylthiophene S-oxide 96 and tetracyclone 97 with benzo[b]thiophene S,S-dioxide (101). Tetracyclone 97 gives pyrone 105 as side product [59, 60].
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Again, cycloaddition reactions of purified thiophene S-oxides can be used to prepare multifunctionalized arenes such as cyclophanes (Scheme 25) [25]. Nakayama et al. [61] have used thiophene S-oxides to prepare sterically over freighted anthraquinones. Thiemann et al. [62] used halogenated thiophene S-oxides, albeit prepared in-situ to synthesize halogenated anthraquinones, which can easily be transformed further to arylated anthraquinones [63, 64]. The cycloaddition reactions of purified thiophene S-oxides can be combined with other transformations in one pot, such as with Wittig olefination reactions (Scheme 20) [55].
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Scheme 25.
Multifunctionalized cyclophanes 108 by cycloaddition of thiophenophane S-oxides 106.
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Not all thiophene S-oxides undergo cycloaddition reactions with alkynes or alkenes. In general, appreciable reaction volume is needed to allow for the forming sulfoxy-bridge in the primary cycloadducts and, in some cases, of the subsequent extrusion of SO. Also, when considerable strain is associated with the thiophene S-oxides and/or the cycloadducts, reactions other than cycloadditions can occur. Thus, strained thiophenophane S-oxide 110 does not undergo a cycloaddition with 98, but undergoes a rearrangement leading to oxygen insertion into the ring with concomitant extrusion of sulfur, leading to furanophane 111 (Scheme 26) [25]. Fujihara et al. were able to prepare the thiacalixarene S-oxide 112; again, the thiacalixarene S-oxide did not undergo a cycloaddition reaction with alkyne 113, but rather formed the thiophene-S,C-sulfonium ylide 114 (Scheme 27) [65].
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Scheme 26.
[2.2]Metathiophenophane S-oxide 109 does not undergo cycloaddition but rearranges to [2.2]furanophane 111.
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Scheme 27.
Thiacalixarene S-oxide 112 reacts with dimethyl acetylenedicarboxylate (113) to the thiacalixarene S,C-ylide 114.
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Thiophene S-oxides as cyclic dienes undergo hetero-Diels-Alder reactions, also (Scheme 28). Thus, Nakayama et al. could establish that 3,4-bis-tert-butylthiophene S-oxide 73 reacts with thioaldehydes 115/117 and thioketones 115, generated in-situ to give 2,7-dithiabicyclo[2.2.1]hept-5-ene 7-oxides 116 and 118 [66] (Scheme 28). The cycloadducts are endo-products as ascertained by X-ray crystallography and 1H NMR spectroscopy. Thiobenzophenone could be reacted with good yield; however, here two isomeric products are produced, the major product originating from the syn-π-face while the lesser product from the anti-π-face cycloaddition.
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Scheme 28.
Hetero-Diels-Alder reactions of 3,4-bis-tert-butylthiophene S-oxide (73).
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Finally, 73 reacts with carbonyl cyanide [121, CO(CN)2], created in-situ by oxidation of tetracyanoethylene oxide (119, TCNO) with thiophene S-oxide 73, in hetero-Diels-Alder fashion to give 122 [67] (Scheme 29).
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Scheme 29.
Reaction of 3,4-tert-butylthiophene S-oxide (73) with tetracyanoethylene oxide (119, TCNO) and hetero-Diels Alder reaction to carbonyl cyanide (121).
\n
Nakayama et al. have calculated that the cycloadditions of the thiophene S-oxides are inverse electron demand reactions [53]. All of the above cycloaddition reactions are highly stereoselective, regardless whether the thiophene S-oxide is prepared and used in-situ or an isolated thiophene S-oxide is used. It is known that the thiophene S-oxides invert at the sulfur and inversion barriers have been calculated and measured experimentally for a number of these compounds [32, 68, 69]. Nevertheless, the sulfoxy group in the 7-thiabicyclo[2.2.1]heptene S-oxide systems is configurational stable. All the cycloadducts are endo-products. In the cases where Lewis acids are used at low temperatures, this in itself is not surprising as it is known that low temperatures kinetically controlled cycloadducts are favored. Moreover, it has been stated that Lewis acid catalysis increases the extent of endo-addition in Diels-Alder reactions [70, 71]. The cycloadditions are seen to have syn-π-facial in that the dienophile adds syn to the oxygen. This means that the lone pair of the sulfur is directed towards the side of the newly formed double bond of the cycloadduct. A number of explanations have been given for the π-facial selectivity. Thus, Nakayama et al. rationalized that in the transition state less geometric change of the SO function would be required to reach the syn- rather than the anti-transition state geometry [53]. Also, a destabilizing interaction between the HOMO of the dienophile and the sulfur lone pair was noted in the anti-transition state [72]. The π-facial selectivity has also been explained by the Cieplak effect [73, 74, 75]. This effect was first proposed to account for the directing effect of remote substituents in addition reactions to substituted cyclohexanones. A large number of experimental observations in Diels-Alder reactions of dienophiles with 5-substituted cyclopentadienes have shown that the dienophiles will approach anti to the antiperiplanar σ-bond that is the better donor at the 5-position of the cyclopentadiene [76]. This σ-bond will best stabilize the σ-bonds formed in the transition state. Cycloadditions to thiophene S-monoxides have been predicted to occur anti to the lone electron-pair on sulfur, which is the better hyper-conjugative donor when compared to the oxygen of the sulfoxy-moiety. The lone pair electron orbital at the sulfur will stabilize the vacant σ*-orbitals of the developing incipient σ-bonds better than any orbital associated with the oxygen of the sulfoxy moiety [77] (Figure 5). This would be even more so, when the oxygen of the sulfoxy-unit is complexed by BF3·Et2O.
\n
Figure 5.
Transition state 123 preferred over transition state 124.
\n
Based on DFT computational studies, Houk et al. [23] showed that the ground state geometry of a thiophene S-oxide already resembles the molecule in its syn transition state. This distortion from planarity of the molecule minimizes its potential antiaromaticity which would result from a hyperconjugative effect by an overlap of σ*S〓O with the π-system (see also above/below) [23] (Figure 6).
\n
Figure 6.
Structural feature of thiophene S-oxide 160.
\n
\n
\n
4.2. Further cycloaddition reactions
\n
When heated with 2-methylene-1,3-dimethylimidazoline (125), 3,4-bis(tert-butyl)thiophene S-oxide 73 undergoes a [4π + 4π]-cycloaddition to the head-to-head dimer 126 (Scheme 30) [78]. Oxidation of the two sulfoxy bridges to sulfone 127 with dimethyldioxirane as oxidant is followed by thermally driven extrusions of the SO2 bridges in 127 and gives 1,2,5,6-tetra(tert-butyl)octatetraene 128 [79] (Scheme 30).
\n
Scheme 30.
[4π + 4π]-cycloaddition of thiophene S-oxide (73) to dimer 126.
\n
Thiophene S-oxides react as enes in 1,3-dipolar cycloaddition reactions. Thus, 3,4-bis-tert-butylthiophene S-oxide (73) reacts with pyrroline N-oxide (129) to give cycloadduct 130 (Scheme 31) [80]. Nakayama et al. could show that 73 reacts with nitrile oxides, diazomethane, nitrile imides, nitrones, and azomethine ylides in syn-π-facial fashion [80].
\n
Scheme 31.
[3 + 2]-cycloaddition of thiophene S-oxide (73) with pyrroline N-oxide (129) as 1,3-dipole.
\n
\n
\n
4.3. Additions to thiophene S-oxides and other reactions
\n
1,4-Additions are known for both 3,4-disubstituted and 2,5-disubstituted thiophene S-oxides [81, 82, 83]. Thus, bromine adds cis to both 3,4-bis-tert-butylthiophene S-oxide (73) [81] and 2,5-bis-trimethylsilylthiophene S-oxide (134) [82] to give the 2,5-dibromo-2,5-dihydrothiophene S-oxide derivatives 131 and 135 (Scheme 32). 3,4-Bis-tert-butylthiophene S,S-dioxide (132) undergoes cis-1,4-bromination, too [81] (Scheme 32). Also, alcohols and mercaptans have been submitted successfully to 1,4-additions with 3,4-bis-tert-butyl thiophene S-oxide (73) (Scheme 33) [83]. Interestingly, disulfur dichloride (S2Cl2) could be added to thiophene S-oxide 73, leading to the rapid formation of adduct 137 (Scheme 34) [84]. 137, however, is not stable and transforms into 138. 138 can be obtained with a 98% yield, when 137 is treated with aq. NaHCO3 (Scheme 34) [84].
\n
Scheme 32.
Bromination of thiophene S-oxides 73 and 134 and thiophene S,S-dioxide 132.
\n
Scheme 33.
Addition of methylthiolate to thiophene S-oxide (73).
\n
Scheme 34.
Addition of disulfur dichloride (S2Cl2) to thiophene S-oxide 73.
\n
The sulfoxy group in thiophene S-oxide can be transformed into a sulfilimine or a sulfoximine moiety [85, 86, 87]. When thiophene S-oxide 73 is reacted with trifluoroacetic acid anhydride or triflic anhydride at −78°C, a mixture of sulfonium salt 139 and sulfurane 140 forms, which can be reacted with p-toluenesulfonamide (141) to provide, as the reaction mixture warms to room temperature, sulfilimine 142 (Scheme 35) [85, 86]. Sulfoximine 145 could be prepared by action of N-[(p-tolylsulfonyl)imino]phenyliodinane (TsN〓IPh, 144) on 2,4-bis-tert-butylthiophene S-oxide (143) in the presence of Cu(CH3CN)4PF6 as catalyst. Further reaction of 145 with H2SO4 leads to N-unsubstituted sulfoximine 146 (Scheme 36) [86].
\n
Scheme 35.
Preparation of thiophene S-imide 142 from thiophene S-oxide 73.
\n
Scheme 36.
Thiophene sulfoximines 145 and 146 from thiophene S-oxide 143.
\n
\n
\n
4.4. Photochemistry of thiophene S-oxides
\n
The photochemical deoxygenation of dibenzothiophene S-oxides has been studied for quite some time [88, 89, 90, 91] and has been found to proceed via the release of ground state atomic oxygen [O(3P)] upon photoirradiation (Scheme 37). Thiophene S-oxides deoxygenate photochemically as well. Nevertheless, the photochemistry of thiophene S-oxides is intrinsically more complex than that of dibenzothiophene S-oxides, often providing a mixture of products, depending on the substitution pattern of the photoirradiated thiophene S-oxide. The photolysis of 2,5-bis(trimethylsilyl)thiophene S-oxide (134) leads exclusively to deoxygenation to produce 2,5-trimethylsilylthiophene (149) (Scheme 38). Otherwise, in those cases, where the thiophene S-oxide does not exhibit a CH3 substituent on the ring system, furans are often the main products along with (deoxygenated) thiophenes (Scheme 39). This has been noted with phenyl-substituted (96, 160) and tert-butyl substituted thiophene S-oxides (73, 143, 153) as well as with 3,4-dibenzylthiophene S-oxide (158) (Scheme 40) [92, 93, 94, 95]. Different mechanisms have been forwarded for this photochemical formation of furans. A viable mechanism involves a cyclic oxathiin, where the first step within the photochemical reaction is initiated by the homolytic ring cleavage α to the sulfoxy group [92, 93, 94]. A rearrangement of thiophene S-oxides to produce furans can also proceed thermally as found by Thiemann et al. [18] in the transformation of thiophenophane S-oxide 110 to furanophane 111 (Scheme 26) and by Mansuy, Dansette et al. in their oxidation of 2,5-diphenylthiophene (162) with H2O2/CF3CO2H to 2,5-diphenylthiophene S-oxide (163), where an appreciable amount of furan 164 was formed as side-product [46] (Scheme 41). In the case of methyl substituted thiophene S-oxides, hydroxyl-alkylthiophenes such as 166 and follow-up products such as ether 167 have been isolated as photoproducts [96] (Scheme 42).
\n
Scheme 37.
Photodeoxygenation of dibenzothiophene S-oxide (147).
\n
Scheme 38.
Photolysis of 2,5-bis(trimethylsilyl)thiophene S-oxide (134).
\n
Scheme 39.
Photolysis of tetraphenylthiophene S-oxide (96).
\n
Scheme 40.
Photolysis of 2,4-bis(tert-butyl)-, 2,5-bis(tert-butyl), 3,4-bis(tert-butyl), 3,4-dibenzyl-, and 2,5-diphenylthiophene S-oxide (143, 153, 73, 158, and 160).
\n
Scheme 41.
Formation of furan 163 in the oxidation of 2,5-diphenylthiophene (162).
\n
Scheme 42.
Photolysis of 3,4-dibenzyl-2,5-dimethylthiophene S-oxide (165).
\n
\n
\n
4.5. Electrochemistry of thiophene S-oxides
\n
Thiophene S-oxides such as 164 and 167 show well-defined, chemically irreversible CV reduction waves, where two reduction processes seem to compete. In the presence of a proton donor, the reduction waves experience a significant shift to more positive potentials, although the reduction potential is still dependent on the substitution pattern of the thiophene S-oxides [96]. In the presence of a proton donor such as benzoic acid at higher concentrations, the reduction of a thiophene S-oxide such as of 167 becomes a straightforward two proton—two electron reduction process to the corresponding thiophene [96]. Bulk electrolysis of thiophene S-oxides in presence of 10-fold excess of benzoic acid has been carried out and have led to the corresponding thiophenes in up to 90% isolated yield (Scheme 43) [96]. Also, thiophene S-oxides show oxidative electrochemistry at platinum in MeCN/Bu4NPF6 [97]. The electrochemical oxidation of tetraphenylthiophene S-oxide under the above conditions leads mainly to the formation of diphenylacylstilbene [98]. Here, more effort needs to be invested to identify the electro-oxidative transformations of other thiophene S-oxides.
\n
Scheme 43.
Electrochemical reduction of 3,4-dibromo-2,5-dimethylthiophene S-oxide (167) in the presence of 10 eq. benzoic acid.
\n
\n
\n
4.6. Structural studies on thiophene S-oxides
\n
In 1990, Rauchfuss et al. published an X-ray crystal structure of the tetramethylthiophene S-oxide rhodium complex 56 [34]. The first X-ray single crystal structure determination of a non-liganded thiophene S-oxide was carried out by Meier-Brocks and Weiss on tetraphenylthiophene S-oxide. The crystal, however, showed some disorder, and only limited information could be gleaned from it [39]. In 1995, Mansuy et al. carried out an X-ray crystal structural analysis of 2,5-diphenylthiophene S-oxide (160) [46, 47], where the structure of 160 was compared to 2,5-diphenylthiophene (162) and 2,5-diphenylthiophene S,S-dioxide (169). The S▬O bond in the thiophene S-oxide was found with 1.484(3) Å to be appreciably longer than those of the thiophene S,S-dioxide with 1.418(5) Å and 1.427(5) Å, respectively [47]. The ring system of the thiophene S,S-dioxide 169 was found to be absolutely planar, while thiophene S-oxide 160 was found to be puckered, with the sulfur lying outside the plane constructed by the four ring carbons by 0.278 Å, and the sulfoxy oxygen lying outside of the plane on the side opposite to sulfur, located by 0.746 Å away from the plane. Previously, this non-planarity of thiophene S-oxides had been predicted by MNDO [99] and ab-initio calculations [100] of the parent thiophene S-oxide itself and dibenzothiophene S-oxide. A more pronounced alteration between double and single C▬C bond was found in thiophene S-oxide 160 in comparison to diphenylthiophene [47]. In probing the aromaticity of thiophene S-oxide 160, it can be seen that apart from its non-planarity, it exhibits relatively large bond order alternations [C(2)▬C(3) 2.11; C(3)▬C(4) 1.23, C(2)/C(5)▬S 1.11; for comparison, the bond orders in 162: C(2)▬C(3) 1.94; C(3)▬C(4) 1.46; C(2)/C(5) 1.53]. The corresponding 2,5-diphenylthiophene S,S-dioxide, though features even larger bond alternations than 160 [47]. An approach for an assessment of aromaticity is the A index as defined by Julg and François [101], which evaluates aromaticity in respect to bond alternation and bond delocalization in ring systems. Here, benzene as the aromatic system par excellence, has an A index of 1, the thiophene system in 2,5-diphenylthiophene has an A index of 0.99, the 5-membered ring system in 2,5-diphenylthiophene S-oxide’s A index is calculated at 0.79, and the parent thiophene S-oxide A index lies at 0.69 ([47], see also [102]).
\n
Subsequently, further X-ray crystal structure analyses were carried out on thiophene S-oxide, such as on 2,5-bis(diphenylmethylsilyl)thiophene S-oxide [45], 3,4-bis-tert-butylthiophene S-oxide (73) [43], (1,1,7,7-tetraethyl-3,3,5,5-tetramethyl-s-hydrindacen-4-yl)thiophene S-oxide [68], 1,3-bis(thien-2yl)-4,5,6,7-tetrahydrobenzo[c]thiophene S-oxide [40], and the sexithiophene (170) (Figure 7), where two of the thienyl units were oxidized to sulfoxides [103]. As the thiophene S-oxides are not planar, they invert at the sulfur with different substituents at the C2/C5 positions leading to different barriers of inversion, which have in part been determined experimentally [32, 68, 69]. Structural features of thiophene S-oxides and thiophene S,S-dioxides have been reviewed before [104].
\n
Figure 7.
Oligothiophene S-oxide 170.
\n
\n
\n
4.7. Oligomers and polymers incorporating thiophene S-oxide units
\n
Oligothiophenes and polythiophenes are being studied as advanced materials with interesting electronic and nonlinear optical properties [105] with applications in photovoltaic cells [106] and field effect transistors (FETs) [107], among others. It has been noted that oxidation of thienyl-units in oligothiophenes and polythiophenes leads to a lowering of energy gaps, to greater electron affinities, and to greater ionization energies [103, 108, 109]. The introduction of thienyl-S,S-dioxides into oligothiophenes often leads to solubility problems of the materials and often leads to a noticeable increase of oxidation potentials. Therefore, there has been a recent interest in incorporating thienyl S-oxide units in oligo- and polythiophenes with the aim of greater solubility and smaller oxidation potentials and narrower energy gaps with electron-affinities similar to thienyl S,S-dioxides [103].
\n
A number of synthetic approaches exist towards the preparation of oligothiophenes with thienyl S-oxide units. Oxidation of a pre-prepared oligo- or polythiophene is more difficult to achieve and leads to modest yield [110]. However, two strategies can be seen as promising. One is the transformation of polyarylene-alkynes 171 via oligozirconacyclopentadienes 172 to polythiophene S-oxides 173, where the zirconacyclopentadienes are reacted with SO2 [41] (Scheme 44). The other takes advantage of the fact that certain thiophene S-oxides such as 2-bromo-3,4-diphenyl-thiophene S-oxide (175) are stable enough to be subjected to C▬C cross-coupling reactions and subsequent halogenation reactions with N-bromosuccinimide (NBS), leading to sequences as shown in Scheme 45 [103]. Already, an FET has been synthesized with a thienyl-thienyl S-oxide polymer [103]. Also, larger π-conjugated ring systems with a thienyl S-oxide unit such as 179 have attracted some attention because of their electronic and optical properties (Figure 8) [111]. As a drawback, it may be noted that thienyl S-oxides in oligomers and polymers would not be stable towards UV radiation as opposed to thienyl S,S-dioxides [112, 113].
\n
Scheme 44.
Preparation with oligomer 173 via zirconacyclopentadiene 172.
\n
Scheme 45.
Preparation of thienyl S-oxide containing oligomers 170 and 178 by Pd(0) Suzuki and Stille cross-coupling reactions.
4.8. Thiophene S-oxides as metabolites in the enzymatic oxidation of thiophenes
\n
Thiophenes have been known to have toxic effects [114, 115]. The understanding of the mechanism leading to the toxicity of thiophenes is of importance, as a number of drugs such as tienilic acid (180), ticlopidine (182), methapyrilene (183), thenalidine (184), tenoxicam (185), cephaloridine (186), suprofen (187), and clopidogrel (188) carry thienyl units, where some of the drugs have been taken off the market (Figure 9). Already in 1990, it was shown that hepatic cytochrome P450 mediated oxidation of the thienyl-containing tienilic acid (180) led to electrophilic metabolites that bind to hepatic proteins [116, 117]. Oxidative metabolism of thiophenes in rats involves thiophene S-oxides [118, 119, 120]. It has been found [119, 121] that rats administered with thiophene (1) in corn oil showed dihydrothiophene S-oxide 191 in their urine as a major metabolite [119] (Scheme 46). This metabolite was assumed to stem from the addition of glutathione (189) to a reactive intermediate thiophene S-oxide 3 (Scheme 46). Previously, it had been shown that rat liver microsomal cytochrome P450 oxidizes 3-aroylthiophene 181, a regioisomer of tienilic acid (180), to aroylthiophene S-oxide 192, which in the presence of mercaptoethanol (193) transformed into dihydrothiophene S-oxide 194 [121] (Scheme 47). Also, 181 was oxidized by clofibrate induced rat liver microsomes to S-oxide 191, which was then trapped as a Diels Alder product with maleimides, for example as 195 [120] (Scheme 48).
\n
Figure 9.
Thiophene-containing pharmaceuticals.
\n
Scheme 46.
Cytochrome P450 mediated transformation of thiophene 1 to adduct 191.
\n
Scheme 47.
Transformation of tienilic acid regioisomer 181 to thiophene S-oxide and its addition of mercaptoethanol (193).
\n
Scheme 48.
Cycloaddition of the thiophene S-oxide derivative of 181 to maleimide.
\n
The oxidation of 2-(4-chlorobenzoyl)thiophene (196), a molecule in structure close to tienilic acid, by H2O2 in the presence of trifluoroacetic acid (TFA) and by m-CPBA, BF3·Et2O, both in CH2Cl2, gives sesquioxides 198–200 that clearly indicate that a thiophene S-oxide structure 197 is formed as an intermediate [122] (Scheme 49). Nevertheless, the oxidation of thiophene (1) itself with H2O2 in the presence of TFA produces apart from sesquioxides 6–8 thiophen-2-one (thiolactone 202). Thiophen-2-one (202) most likely is produced through thiophene-epoxide (201) [23] (Scheme 50). Thiophen-2-one (202) is in equilibrium with 2-hydroxythiophene (202). There is one report of a Pummerer-like rearrangement reaction that leads from the purified and isolated thiophene S-oxide 134 to thiophen-2-one (thiolactone 202) [123] (Scheme 51). Still, the current understanding is that the thiophene S-oxide intermediates formed in vivo do not lead to a 2-hydroxythiophene (203) [124] (Scheme 52), so that two separate mechanisms may exist for the cytochrome P450 2C9 (CYP2C9) mediated oxidation of thiophenes. In this regard, Dansette et al. [119] showed that CYP450s may catalyze both the reaction of thiophenes to thiophene S-oxide and to thiophene epoxides [125].
\n
Scheme 49.
Formation of sequioxides 198–200 by dimerization of thiophene S-oxide 197.
\n
Scheme 50.
Reaction of thiophene (1) leads via thiophene S-oxide (3) to sesquioxides 7–9 and in a separate pathway via thiophene epoxide 201 to thiolactone 202 and thus to 2-hydroxythiophene (203).
\n
Scheme 51.
Pummerer reaction of thiophene S-oxide 134 to thiolactone 202.
\n
Scheme 52.
Cytochrome P450 mediated oxidation of thiophene may lead to two pathways, one through thiophene S-oxide 3, the other through thiophene epoxide 201.
\n
Also, the investigation of the metabolism of other thienyl-containing pharmaceuticals show that potentially both mechanisms, epoxidation of the thiophene-unit and oxidation of the thiophene-unit to thiophene S-oxide, operate concurrently. As to the thiophene S-oxide pathway, Shimizu et al. in their investigation of metabolites ticlopidine (182) in rats found both the glutathione conjugate of ticlopidine S-oxide 205 and the dimeric ticlopidine S-oxide cycloadduct 206 (Figure 10) [126, 127]. The structures could be identified by mass spectrometry, and 1H and 13C NMR spectrometry. Medower et al. have noted that cytochrome P450 mediated oxidation of cancer drug OSI-930 (207) leads to GSH conjugate 209, derived from OSI-930 S-oxide (208), as recognized by mass spectrometry (Scheme 53) [128].
\n
Figure 10.
Metabolites of ticlopidine that derive from a ticlopidine S-oxide intermediate.
\n
Scheme 53.
In vivo oxidation of anticancer drug OSI-930 (207) to OSI-930 sulfoxide (208) and addition of glutathione (GSH) to provide identified metabolite 209.
\n
Lastly, both possible metabolic pathways of thiophenes, via thiophene S-oxides and via thiophene epoxides, have been examined as to their energy profiles using density functional theory [129]. It was found that the formation of the thiophene epoxide (−23.24 kcal/mol) is more exothermic than the formation of the thiophene S-oxide (−8.08 kcal/mol) [129]. Also, the formation of thiophene epoxide seems kinetically favored [129]. Both possible metabolites, thiophene S-oxide and thiophene epoxide, are highly electrophilic, leading to bond formation with nucleophiles such as with amino acids, leading to a mechanism-based inactivation (MBI) of cytochrome P450.
\n
\n
\n
\n
5. Conclusion
\n
Since the first unverified isolation of a thiophene S-oxide a little more than 50 years ago, research on thiophene S-oxides has reached a milestone. Due to mainly two synthetic routes, the controlled oxidation of thiophenes in presence of a Lewis- or proton acid and the reaction of zirconacyclopentadienes with thionyl chloride, a number of thiophene S-oxides have now become readily accessible. Thiophene S-oxides are noted to be reactive dienes in Diels-Alder type cycloadditions, where they react equally well with electron-poor and electron-rich dienophiles. Thiophene S-oxides can be stabilized by sterically exacting substituents. Then, they exhibit sufficient stability to be submitted to Pd(0)-catalyzed cross-coupling reactions without deoxygenation.
\n
This leads to the possibility of preparing aryl-oligomers with thiophene-S-oxide subunits. By comparing oligothiophenes and oligomers with thiophene S,S-dioxide subunits, oligomers with thiophene S-oxide subunits exhibit smaller oxidation potentials and narrower energy gaps with electron-affinities greater than oligothiophenes and similar to thiophene S,S-dioxides. Nevertheless, thiophene S-oxides are not stable photochemically, but deoxygenate to the corresponding thiophenes or transform to furans by photochemical rearrangement.
\n
Thiophene S-oxides have been found to act as intermediates in the cytochrome P540 mediated, oxidative metabolism of thiophene-containing compounds, including a number of important thiophene containing pharmaceuticals. Addition of nucleophiles in vivo leads to mechanism based inhibition (MBI) and to toxic side effects of the thiophenes, including nephrotoxicity.
\n
\n\n',keywords:"thiophenes, selective oxidation, cycloaddition, functionalized arenes, drug metabolites",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/62502.pdf",chapterXML:"https://mts.intechopen.com/source/xml/62502.xml",downloadPdfUrl:"/chapter/pdf-download/62502",previewPdfUrl:"/chapter/pdf-preview/62502",totalDownloads:684,totalViews:250,totalCrossrefCites:0,dateSubmitted:"January 12th 2018",dateReviewed:"May 23rd 2018",datePrePublished:"November 5th 2018",datePublished:"February 20th 2019",dateFinished:"July 6th 2018",readingETA:"0",abstract:"Thiophene S-oxides constitute a class of molecules that have been studied in more detail only recently. Their existence as intermediates in the peracid mediated oxidation of thiophenes to thiophene S,S-dioxides, however, has been known over some time. Over the last 20 years, a larger number of thiophene S-oxides have been prepared and isolated in pure form. Thiophene S-oxides have been found to be good dienes in [4 + 2]-cycloaddition reactions, where they react with electron-poor, electron-neutral and electron-rich dienophiles with high syn π-facial stereoselectivity. Thiophene S-oxides have been found to be metabolites of thienyl-containing pharmaceuticals such as the anti-platelet drugs ticlopidine and clopidogrel. The chapter gives an overview of the preparation and reactivity of this class of compounds.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/62502",risUrl:"/chapter/ris/62502",signatures:"Thies Thiemann",book:{id:"6797",title:"Chalcogen Chemistry",subtitle:null,fullTitle:"Chalcogen Chemistry",slug:"chalcogen-chemistry",publishedDate:"February 20th 2019",bookSignature:"Peter Papoh Ndibewu",coverURL:"https://cdn.intechopen.com/books/images_new/6797.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"87629",title:"Prof.",name:"Peter",middleName:"Papoh",surname:"Ndibewu",slug:"peter-ndibewu",fullName:"Peter Ndibewu"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"199012",title:"Prof.",name:"Thies",middleName:null,surname:"Thiemann",fullName:"Thies Thiemann",slug:"thies-thiemann",email:"thiesthiemann@medjchem.com",position:null,institution:{name:"United Arab Emirates University",institutionURL:null,country:{name:"United Arab Emirates"}}}],sections:[{id:"sec_1",title:"1. Early history of oxidation reactions of thiophenes: cycloaddition reactions of thiophene S-oxides prepared in situ in absence of Lewis acids",level:"1"},{id:"sec_2",title:"2. Cycloaddition reactions of thiophene S-oxide prepared in situ in the presence of Lewis acids: thiophene S-oxides are isolated",level:"1"},{id:"sec_3",title:"3. Preparation and isolation of pure thiophene S-oxides",level:"1"},{id:"sec_4",title:"4. Reactions of thiophene S-oxides",level:"1"},{id:"sec_4_2",title:"4.1. [4 + 2]-cycloaddition reactions",level:"2"},{id:"sec_5_2",title:"4.2. Further cycloaddition reactions",level:"2"},{id:"sec_6_2",title:"4.3. Additions to thiophene S-oxides and other reactions",level:"2"},{id:"sec_7_2",title:"4.4. Photochemistry of thiophene S-oxides",level:"2"},{id:"sec_8_2",title:"4.5. Electrochemistry of thiophene S-oxides",level:"2"},{id:"sec_9_2",title:"4.6. Structural studies on thiophene S-oxides",level:"2"},{id:"sec_10_2",title:"4.7. Oligomers and polymers incorporating thiophene S-oxide units",level:"2"},{id:"sec_11_2",title:"4.8. Thiophene S-oxides as metabolites in the enzymatic oxidation of thiophenes",level:"2"},{id:"sec_13",title:"5. Conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'Lanfry M. Sur les oxythiophènes. Comptes Rendus. 1911;153:73-76\n'},{id:"B2",body:'Lanfry M. ur les oxy-b-méthylthiophènes. Comptes Rendus. 1911;153:821-822\n'},{id:"B3",body:'Bailey WJ, Cummins EW. Cyclic dienes. III. Synthesis of thiophene 1-dioxide. Journal of the American Chemical Society. 1954;76:1932-1936\n'},{id:"B4",body:'Bailey WJ, Cummins EW. Cyclic dienes. IV. The dimerization of thiophene 1-dioxide. Journal of the American Chemical Society. 1954;76:1936-1940\n'},{id:"B5",body:'Nagasawa H, Sugihara Y, Ishii A, Nakayama J. Thiophene 1,1-dioxide: Synthesis, isolation, and properties. Bulletin of the Chemical Society of Japan. 1999;72:1919-1926\n'},{id:"B6",body:'Benders PH, Reinhoudt DN, Trompenaars WP. 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Journal of Physical Organic Chemistry. 2000;13:648-653\n'},{id:"B97",body:'Valcarel JI, Walton DJ, Fujii H, Thiemann T, Tanaka Y, Mataka S, Mason TJ, Lorimer JP. The sonoelectrooxidation of thiophene S-oxides. Ultrasonics Sonochemistry. 2004;11:227-232\n'},{id:"B98",body:'Iniesta J, Alcock H, Walton DJ, Watanabe M, Mataka S, Thiemann T. Electrochemical oxidation of tetracyclones and tetraphenylthiophene S-oxide. Electrochimica Acta. 2006;51:5682-5690\n'},{id:"B99",body:'Hashmall JA, Horak V, Khoo LE, Quicksall CO, Sun MK. Molecular structure of selected S-methylthiophenium tetrafluoroborates and dibenzothiophene S-oxide. Journal of the American Chemical Society. 1981;103:289-295\n'},{id:"B100",body:'Rozas I. Comparative study of aromaticity in five-membered rings containing S, SO, and SO2 groups. Journal of Physical Organic Chemistry. 1992;5:74-82\n'},{id:"B101",body:'Julg A, François P. Recherches sur la géométrie de quelques hydrocarbures non-alternants: Son influence sur les énergies de transition, une nouvelle définition de l\'aromaticité. Theoretica Chimica Acta. 1967;7:249-259\n'},{id:"B102",body:'Holloczki O, Nyulaszi L. Analogy between sulfuryl and phosphino groups: The aromaticity of thiophene-oxide. Structural Chemistry. 2011;22:1385-1392\n'},{id:"B103",body:'Di Maria F, Zangoli M, Palama IE, Fabiano E, Zanelli A, Monari M, Perinot A, Caironi M, Maiorano V, Maggiore A, Pugliese M, Salatelli E, Gigli G, Viola I, Barbarella G. Improving the property-function tuning range of thiophene materials via facile synthesis of oligo/polythiophene-S-oxides/oligo/polythiophene-S,S-dioxides. Advanced Functional Materials. 2016;26:6970-6984\n'},{id:"B104",body:'Lukevics E, Arsenyan P, Belyakov S, Pudova O. Molecular structure of thiophene 1,1-dioxides, thiophene S-oxides and their derivatives. 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Biochemical and Biophysical Research Communications. 1992;186:1624-1630\n'},{id:"B120",body:'Dansette PM, Thebault S, Durand-Gasselin L, Mansuy D. Reactive metabolites of thiophenic compounds: A new trapping method for thiophene sulfoxides. Drug Metabolism Reviews. 2010;32(Suppl. 1):233-234 presentation at ISSX, Istanbul\n'},{id:"B121",body:'Mansuy D, Valadon P, Erdelmeier I, Lopez-Garcia P, Amar C, Girault JP, Dansette PM. Thiophene S-oxides as new reactive metabolites: Formation by cytochrome P450 dependent oxidation and reaction with nucleophiles. Journal of the American Chemical Society. 1991;113:7825-7826\n'},{id:"B122",body:'Ho MT, Treiber A, Dansette PM. Oxidation of 2-(4-chlorobenzoyl)thiophene into 1-oxide Diels-Alder dimers, sesquioxide and a sulfone-water adduct. Tetrahedron Letters. 1998;39:5049-5052\n'},{id:"B123",body:'Sato S, Zhang S-Z, Furukawa N. The Pummerer-like reaction of 2,5-bis(trimethylsilyl)thiophene S-oxide with trifluoroacetic anhydride: Intermediary formation of sulfurane. Heteroatom Chemistry. 2001;15:444-450\n'},{id:"B124",body:'Rademacher PM, Woods CM, Huang QB, Szklarz GD, Nelson SD. Differential oxidation of two thiophene-containing regioisomers to reactive metabolites by cytochrome P450 2C9. Chemical Research in Toxicology. 2012;25:895-903\n'},{id:"B125",body:'Dansette PM, Bertho G, Mansuy D. First evidence that cytochrome P450 may catalyze both S-oxidation and epoxidation of thiophene derivatives. Biochemical and Biophysical Research Communications. 2005;338:450-455\n'},{id:"B126",body:'Shimizu S, Atsumi R, Nakazawa T, Fujimaki Y, Sudo K, Okazaki O. Metabolism of ticlopidine in rats: Identification of the main biliary metabolite as a glutathione conjugate of ticlopidine S-oxide. Drug Metabolism and Disposition. 2009;37:1904-1915\n'},{id:"B127",body:'Ha-Duong NT, Dijols S, Macherey AC, Goldstein JA, Dansette PM, Mansuy D. Ticlopidine as a selective mechanism-based inhibitor of human cytochrome P450 2C19. Biochemistry. 2001;40:12112-12122\n'},{id:"B128",body:'Medower C, Wen L, Johnson WW. Cytochrome P450 oxidation of the thiophene-containing anticancer drug 3-[(quinolin-4-ylmethyl)-amino]-thiophene-2-carboxylic acid (4-trifluoromethoxyphenyl)amide to an electrophilic intermediate. Chemical Research in Toxicology. 2008;21:1570-1577\n'},{id:"B129",body:'Jaladanski CK, Taxak N, Varikoti RA, Bharatam PV. Toxicity originating from thiophene containing drugs: Exploring the mechanism using quantum chemical methods. Chemical Research in Toxicology. 2015;28:2364-2376\n'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Thies Thiemann",address:"thies@uaeu.ac.ae",affiliation:'
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Epithelial ovarian cancer (EOC) has long been considered a heterogeneous disease with respect to histopathology, molecular biology, and clinical outcome. Treatment of ovarian cancer includes a combination of cytoreductive surgery and combination chemotherapy, with platinums and taxanes. Despite initial success, over 75% of patients with advanced disease will relapse around 18 months and the overall 5-year survival is approximately 50%. Cancer cells are known to be under intrinsic oxidative stress, which alters their metabolic activity and reduces apoptosis. Epithelial ovarian cancer has been shown to manifest a persistent pro-oxidant state as evident by the upregulation of several key oxidant enzymes in EOC tissues and cells. 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