Anisotropic Kepler Problem and Critical Level Statistics

The subject of this chapter is quantum chaos (QC), in particular, the QC that occurs in the Anisotropic Kepler Problem (AKP). In QC, one studies a quantum system whose classical counter part is chaotic, and one investigates how the chaotic property in the classical theory shows up its imprints in the quantum theory. To elucidate this quantum-classical correspondence is the first mission of the quantum chaos study. With the advent of nanophysics techniques, QC has become very important concern also at the experimental side; for pioneering works, we refer to conductance fluctuations in quantum dot (Marcus et. al., 1992), magnetoresistance on a superlattice of antidots (Weiss et al., 1991), a cold-atom realization of the kicked top (Chaudhury et al., 2009). Also for an interesting experimental observation of quantum scars of classical orbits (Heller, 1984; 1989), we refer to (Stein & Stöckmann, 1992). Furthermore, quantum chaos study has been developed under far-reaching mutual influences with related areas. In order to explain where our study in AKP stands, let us briefly review a few aspects of quantum chaos study. Let us first consider the random matrix theory (RMT) (Mehta, 2004). It is proposed by (Wigner, 1951) to predict the universal spectral property of complex nucleus, and the mathematical basement is set by (Dyson, 1962), (Dyson & Mehta, 1963) and (Mehta & Dyson, 1963). In RMT, the hamiltonian of the physical system is described by a random matrix in the three basic ensembles. This implies that the intrinsic quantum property of the physical system is determined by the time reversal symmetry and internal symmetry only, and does not depend on the details of the system hamiltonian. If the time-reversal by the operator T is broken in a system, the relevant ensemble is gaussian random ensemble of hermite matrices for the hamiltonian, admitting the invariance of the hamiltonian under the unitary transformation (GUE), with Dyson parameter β = 2. If the T invariance holds with T2 = 1, it is gaussian orthogonal ensemble (GOE) of real symmetric matrices with β = 1 , while with T2 = −1, it is gaussian symplectic ensemble (GSE) of quarternion selfdual matrices with β = 4. It is conjectured by (Bohigas et al., 1984) that, irrespective to the details of the system hamiltonian, the stochastic spectral property of energy levels of an physical system (including AKP) is uniquely described by the relevant RMT ensemble chosen by the above symmetry property Anisotropic Kepler Problem and Critical Level Statistics


Introduction
The subject of this chapter is quantum chaos (QC), in particular, the QC that occurs in the Anisotropic Kepler Problem (AKP).In QC, one studies a quantum system whose classical counter part is chaotic, and one investigates how the chaotic property in the classical theory shows up its imprints in the quantum theory.To elucidate this quantum-classical correspondence is the first mission of the quantum chaos study.With the advent of nanophysics techniques, QC has become very important concern also at the experimental side; for pioneering works, we refer to conductance fluctuations in quantum dot (Marcus et. al., 1992), magnetoresistance on a superlattice of antidots (Weiss et al., 1991), a cold-atom realization of the kicked top (Chaudhury et al., 2009).Also for an interesting experimental observation of quantum scars of classical orbits (Heller, 1984;1989), we refer to (Stein & Stöckmann, 1992).Furthermore, quantum chaos study has been developed under far-reaching mutual influences with related areas.In order to explain where our study in AKP stands, let us briefly review a few aspects of quantum chaos study.Let us first consider the random matrix theory (RMT) (Mehta, 2004).It is proposed by (Wigner, 1951) to predict the universal spectral property of complex nucleus, and the mathematical basement is set by (Dyson, 1962), (Dyson & Mehta, 1963) and (Mehta & Dyson, 1963).In RMT, the hamiltonian of the physical system is described by a random matrix in the three basic ensembles.This implies that the intrinsic quantum property of the physical system is determined by the time reversal symmetry and internal symmetry only, and does not depend on the details of the system hamiltonian.If the time-reversal by the operator T is broken in a system, the relevant ensemble is gaussian random ensemble of hermite matrices for the hamiltonian, admitting the invariance of the hamiltonian under the unitary transformation (GUE), with Dyson parameter β = 2.If the T invariance holds with T 2 = 1, it is gaussian orthogonal ensemble (GOE) of real symmetric matrices with β = 1 , while with T 2 = −1, it is gaussian symplectic ensemble (GSE) of quarternion selfdual matrices with β = 4.It is conjectured by (Bohigas et al., 1984) that, irrespective to the details of the system hamiltonian, the stochastic spectral property of energy levels of an physical system (including AKP) is uniquely described by the relevant RMT ensemble chosen by the above symmetry property

Anisotropic Kepler Problem and Critical Level Statistics
Kazuhiro Kubo and Tokuzo Shimada Department of Physics, School of Science and Technology, Meiji University Japan only, provided that the quantum system is in the ergodic regime (BGS conjecture). 1 This conjecture2 most remarkably asserts universality classes in the whole lots of quantum chaos systems, and has been successful in great many physical systems3 .N o w ,l e t u s m e n t i o n a famous, though ever charming episode; a mathematician Montgomery was introduced to Dyson at a tea time and explained his result on the spacing distribution of the non-trivial zeros of Riemann zeta function.Then, Dyson immediately told that it is just what he knows, the P(s) of GUE!Of course, one should also mention that the Gutzwiller trace formula that enumerates the semi-classically quantum Green function of a classically chaotic system from unstable periodic orbits has so close intriguing correspondence with the trace formula of the Riemann zeta function (Bohigas, 2005).A deep correspondence between the Riemann zeta function and the random matrix theory is shown by the agreement between the Conrey-Ghosh conjecture on the 2k-th continuous moment of ζ(1/2 + it) and Keating-Snaith random matrix calculation.One of the most flourishing areas in physics is Anderson localization (Evers & Mirlin, 2008) which is only understandable as a quantum phenomena from the interfering amplitudes at the metal and insulator transition point.The critical statistics is a target of critical random matrix theories.So much for general review and let us turn to AKP.
In this chapter, we calculate the energy level statistics of AKP, and find that it is described well (over a finite range of mass anisotropy) by the critical random matrix model devised by (García-García & Verbaarschot, 2003) that is also related to the critical level statistics in the Anderson localization.We also investigate the systematic change of the AKP wave functions with the mass anisotropy.In the Anderson localization, the theory predicts that at the mobility edge, the wave function is multi-fractal, and the level statistics indicator Σ 2 (L) should show a linear rise for large L that is also observed by our data.In this way, our study may give a support to the recent notion Anderson localization in quantum chaos proposed by (García-García, 2007;García-García & Wang, 2008).
The AKP is a system of an electron bound to a proton, just like a hydrogen atom, but the electron has an anisotropic mass.This system is experimentally realized by an electron in a doped semiconductor.Gutzwiller, who made the periodic orbit theory by his semi-classical trace formula, chose often AKP for a nice testing ground of QC (Gutzwiller, 1971;1977;1980;1981;1982;1990).In fact, when mass anisotropy is not present, the system is the hydrogen atom (the Kepler problem) that is one of the most well-studied integrable quantum system.By varying the mass anisotropy parameter, the classical system changes the strength of randomness.It is known that the AKP is not KAM system (García-García & Verbaarschot, 2003;Wintgen & Marxer, 1988) that means that the classical phase space, when the mass anisotropy is present, is not a mixture of integrable regions (tori) and ergodic regions.Instead, with the increase of the anisotropy, the system changes its classical phase space structure due to the gradual collapse of tori, via the structure filled by cantori (Zaslavsky et al., 1991) (stochastic web of chaos), finally to that filled by isolated unstable periodic orbits.Thus, AKP is really a nice testing ground that gives us the opportunity to investigate how systematically, along with increasing anisotropy, the quantum feature changes reflecting the change in the classical phase space structure.Any periodic orbits in AKP in the chaotic regime can be uniquely coded by its own Bernoulli sequences and furthermore Gutzwiller found an amazing approximate formula that gives the action of a periodic orbit from its Bernoulli sequences.
In this chapter we aim to view various faces of QC from AKP taking the above mentioned advantages of AKP.
In section 2, we first calculate the energy levels of AKP at various anisotropy following the method developed and first applied to AKP by (Wintgen et al., 1987).In this method, there is a particular parameter ǫ (see ( 6)) and how to choose it at given anisotropy is crucial to guarantee the accuracy of the levels.We use the Sturmian basis just as in (Wintgen et al., 1987) and we present a couple of simple rules to choose the parameter at a given anisotropy.Also, we set up another formulation based on the harmonic oscillator basis.This provides us with a useful check of our results and, in addition, serves as an efficient method for calculating Husimi function that is an important measure to explore the quantum and classical correspondence.
In section 3, we investigate the level statistics in AKP.There is a concrete result by (Wintgen & Marxer, 1988) for the anisotropy parameter γ = 0.8 (only), but, in order to investigate the quantum statistics change according to the the variation of γ, we definitely need reliable eigenvalue set at various anisotropy too.This is why we have performed the eigenvalue analysis from scratch as described in section 2. We now show for the first time how the number variance and the spectral rigidity change their feature with the variation of mass anisotropy in AKP .TheAKPisasystemthatpreservesthetimereversalinvariancewithT 2 = 1 and hence its level statistics is expected to be described by GOE in the ergodic region (γ ≪ 1), while in the vicinity of Kepler limit (γ = 1), it should be Poissonian.We are interested in the physics in the intermediate range.(García-García & Verbaarschot, 2003) showed that the level statistics of AKP at one particular anisotropy is successfully explained by a critical random matrix theory that has one parameter h (temperature in an equivalent model), using the eigenvalue set at γ = 0.8 given by Wintgen only available at that time, provided that h is suitably chosen.We show that the AKP level statistics can be well described by their critical random matrix theory in a finite range of γ in the intermediate region and show that there is a smooth relation between the parameter h and the anisotropy γ (h ∝ e 7.2γ ).
In section 4, we now turn our attention to various interesting features of wave functions of AKP.We discuss firstly the nodal line systematics.Then we investigate the probability densities and find salient scars of periodic orbits.This observation is strengthened by the subsequent study of Husimi functions of AKP calculated by the method developed in subsection 4.3.
In section 5, we conclude after a briefly discussion on the future outlook of our work, especially, on the possibility to consider quantum chaos in AKP in the context of Anderson transition 2. Non-perturbative matrix method for the evaluation of quantum energy levels

WMB method
First we recapitulate the WMB method in the context of AKP (Wintgen et al., 1987;Wintgen & Marxer, 1988) in the units h2 K/m 1 e 2 , m 1 e4 /2h 2 K 2 ,2h 3 K 2 /m 1 e 4 for length, energy, time, respectively(K is the dielectric constant).One uses Sturmian basis {|nℓm } Here n, ℓ, m are radial, azimuthal, magnetic quantum numbers respectively and they are related to the principle quantum number n p by n p = n + ℓ + 1.The parameter λ is introduced for the scaling of r.(In the left hand side the dependence on λ is suppressed for simplicity).Note for the eigen function with n p in the Kepler problem (γ = 1), λ = 2/n p .I nt h eS t u r m i a n basis the Schrödinger equation of the AKP becomes a matrix equation: where the eigenvector Ψ is Col.( nℓm|Ψ ), and the matrix elements are respectively (The ∂ 2 term is somewhat complicated and we refer the reader to (Wintgen et al., 1987)).
Noting that the Coulombic interaction term is diagonal, while ← → Id is not, one exchanges them between right and left to obtain the standard eigenvalue problem.Dividing the whole equation by λ one obtains Note that E/λ in the right-hand side of ( 4) is now (after divided by λ) changed into a parameter ǫ in the left-hand side of (5), namely This ǫ should be fixed at some constant value in the diagonalization of (5).This fixing is an extremely clever way in WMB.We will explain this point shortly below.
The 'hamiltonian' matrix ← → M is symmetric and block diagonal; nonvanishing elements are those with ℓ ′ = ℓ + 2orℓ, and due to the rotational symmetry about z-axis m ′ = m.Therefore one can organize ← → M as a banded and sparse matrix.By solving (5) for the eigenvalues Λ i ≡ 2/λ i the energy levels are in turn determined by E i = ǫλ 2 i .This is the WMB method.In this scheme (5), the eigen function r|Ψ i of the i-th level E i is calculated from the eigen vector Ψ i using the Sturmian basis with parameter λ i =(E i /ǫ) 1/2 .To be explicit one calculates This is in a sharp contrast to (4) where eigenstates of all energy levels must be calculated with a uniquely chosen λ.With the flexibility of optimizing λ i for each Ψ i ,WMBmet h od(5 )isb y far superior to (4).For instance, if γ ≈ 1, ǫ ≈− 1/4 is a good choice since at ǫ = −1/4 the Sturmian basis is already the proper basis for the first order perturbation theory for AKP in 1 − γ. 5 In fact we can accommodate various γ by adjusting ǫ properly.Supposing that we choose another value for the parameter ǫ of the matrix ← → M , the eigenvalues Λ i s will be accordingly changed, but the physical energies E i should be kept unchanged.However in practice we cannot help using a truncated basis.We discuss how to accommodate this subtle problem by adjusting ǫ shortly below.Hereafter we focus our attention to the m = 0 sector that is related to the 2−dimensional classical AKP.Also for definiteness we consider the case of even ℓ case.With these constraints the matrix where N p is the maximal principal quantum number of the Sturmian basis {|nℓm } ,i .e .
We present in Fig. 1 the energy levels of m = 0 sector obtained by diagonalizing the 'hamiltonian' ← → M in (5) with N = 7921 (N p = 177).The levels are plotted as functions of the scaling parameter ǫ (see ( 5)) for typical values of anisotropy parameter γ, after proper stretch (Bohigas, 2005) using Thomas-Fermi approximation N(E) for the stair case function of AKP (Wintgen & Marxer, 1988) is the level density.Given N(E), the stretched energy levels f i are given by inversion (unfolding map (Bohigas & Giannoni, 1984)); and the average energy spacing f i+1 − f i = 1.The levels must be independent from the scaling parameter ǫ (level curves must be horizontal) in so far as the matrix formalism works properly.(Recall that the WMB method is not a perturbation theory; it should yield exact results if the matrix size N is infinity at any choice of ǫ.)However, at a glance, one finds that the curves are under all over systematic distortion induced by truncating the matrix size.In order to accommodate this, one should select the part of calculated energy levels that satisfy N p =177, N = 7921, ℓ=even, m=0.Sky-and dark-blue mountain-like curves account for necessary conditions 1 and 2 respectively and the bulk under the curves satisfies sampling test (e.g.Fig. 2).In order to make curves, diagonalization is performed for each γ and for thirty values of scaling parameter ǫ ranging from −0.3 to −0.01 with inclement 0.01.The green dashed curve connects the minimum point of f i and hits naturally the peak of the mountain.The level group with level-repulsion property is marked by red; these groups together remarkably overlap the domain for any γ (except the one γ = 0.9 > 8/9).For discussion on this, see subsection 3.1.
the following condition: Condition 1: The level curves must be all horizontal.This is a necessary condition.In each diagram for respective γ the region under the (sky-blue) mountain-shaped curve is the good ǫ-scaling region.
One must consider one more condition.Recall that the spectrum (E i ) is stretched by Thomas-Fermi formula.If this procedure is valid the resultant spectrum ( f i ) must satisfy δ f i ≈ 1 apart from local fluctuations.Let us put this into a quantitative form; 86 Theoretical Concepts of Quantum Mechanics www.intechopen.com Condition 2: The | δ f i n − 1| 0.05 . 6Here n labels one of the 40 groups that the whole levels are divided into by ascending order.The region under the (dark-blue) mountain-shaped curve is the successfully stretched region.One observes a good overall agreement between two-types of regions, which gives a strong support for the celebrated WMB method.The bulk under the mountain-shaped curves in the f i − ǫ plane in Fig. 1 is a candidate region for the study of level statistics satisfying both of necessary conditions.Let us call it the good domain.Now we have to proceed a step forward; the energy levels are under subtle fluctuation and we must examine the level statistics (for each anisotropy γ) in order to understand the quantum AKP theory.Does the good domain maintain a unique level statistics?Note that graphs for the latter two statistics are properly shifted to avoid their overlap.The vertical axis is for (c).
Fig. 2 exhibits a sample of test.Here we have selected three regions (a, b, c), each consists of a thousand levels (5 groups) at a given ǫ.The comparison of the level statistics (Dyson, 1962;Dyson & Mehta, 1963;Mehta, 2004) in this figure succinctly shows that there is no sizable difference of the statistics with respect to any one of the three statistical characteristics; the spacing distribution P(s), the number variance Σ 2 (L) and the spectral rigidity Δ 3 (L).(F o r details of these quantities, see later discussion.)Repeating this test all over the good domain, we have checked that any region in the good main is equally satisfactory for the level statistics.Now we are in the position to propose a prescription to determine the proper ǫ for the eigenvalue analysis on the WMB method; Prescription for ǫ: Choose the ǫ value that gives the maximum height of the good domain; then one can determine the largest number of levels at the given γ and at the given matrix size N.This ǫ is the best.We denote it ǫ * .

Regularities regarding the best choice for ǫ
We have found the following empirical regularities regarding ǫ * .
(i) Fig. 3 shows the relationship between γ and ǫ * .Data points clearly follow a linear line described very well by ǫ * ≃ (−1/4)γ.(ii) Consider the ratio R eff of the number of eigenvalues that satisfy conditions (1) and ( 2) to the number of whole levels N at ǫ * .This is an indicator that tells us how much levels among all are really usable for the investigation of quantum level statistics.Fig. 4 shows plotted R eff as a function of γ.We find it is described very well by Here c = 1 with an error of only one per cent.
We have checked that these observations do not depend on the matrix size N.For instance, the small set N = 1444(N p = 75) shows the same features with the large set N = 7921(N p = 177).This independence assures the following planning of the AKP diagonalization using the WMB method.Using (10) as a rule of thumb, we can first easily estimate the appropriate ǫ * that is appropriate for a given anisotropy γ.Then using (11), we can estimate the necessary matrix size N for obtaining desired number of energy levels.For instance, if one wants to examine the germanium (silicon) levels with γ ≈ 0.05 (0.2), then R eff = 0.23 (0.46) at ǫ * = −0.01(−0.05).
Then, to obtain first say 2000 reliable levels in the ℓ =even and m = 0 sector, one has to choose N ≈ 8800(4400).On the other hand, for γ = 0.8, the appropriate ǫ * now becomes −0.R eff = 0.9. 7Regularities ( 10) and ( 11) are remarkably simple.It is obvious that the ǫ * curve in Fig. 3 starts from ǫ = −1/4 at the Kepler limit (γ = 1) because, as we discussed below (7), the Sturmian basis is already the proper basis for the first order perturbation theory for AKP.This curve presumably represents the effect of truncating the basis in the WMB diagonalization.( 10) and ( 11) are tantalizing.

Application of harmonic oscillator function basis to AKP
Harmonic oscillator function basis is quite useful for quantum chaos study through Husimi function (Husimi, 1940) and we discuss here the application of it to AKP.The formulation is almost parallel to that in subsection 2.1; main difference stems in the interaction terms.Let us introduce the semi-parabolic coordinates μ, ν, φ by The AKP Schrödinger equation in terms of semi-parabolic coordinates is (13) 7 The level statistics of AKP at γ = 0.8 were studied in the seminal paper (Wintgen & Marxer, 1988) but the choice of ǫ was not written there regrettably.Also no other data were published for other γ.
In an earlier article (Wintgen et al., 1987) (the paper of WMB method), the low lying levels of silicon (γ = 0.2079) and germanium (γ = 0.05134) had been presented with the explicit statement of ǫ = −0.01 for both cases.This led us to some confusion at the early stage of this work, because, for γ = 0.8, the proper choice is ǫ ≈− 0.2 much smaller than ǫ = −0.01.Our level statistics results for γ = 0.8 agree with those in (Wintgen & Marxer, 1988) within errors; therefore we believe that ǫ was adjusted properly in (Wintgen & Marxer, 1988).We add that the best choice for Silicon is ǫ = −0.05rather than ǫ = −0.01;this enhances R ef f approximately by factor 3. At these circumstances we revisited the whole level calculation and construct Fig. 1 (plus a dozen diagrams for other anisotropy).
Multiplying by μ 2 + ν 2 and exchanging the Coulombic interaction term and the E term between the right and left hand sides, one obtains Thanks to the semi-parabolic coordinates, the Coulombic singularity has removed (Kustaanheimo & Stiefel, 1965) (for γ = 1).
One has now two of two-dimensional harmonic oscillators coupled by an interaction term (μ 2 + ν 2 )∂ 2 /∂z 2 introduced by the mass anisotropy in the original problem.Let us call the oscillators as μ− and ν− oscillators respectively.The proper basis for each oscillator is the harmonic oscillator basis and for two of them one uses tensor product of these bases.The μ− harmonic basis {|j, m } is defined as eigenstates of the Schrödinger equation for the two-dimensional harmonic oscillator and the polar-coordinate representation of normalized |j, m is given by For the m = 0 sector of AKP, one should set m μ = m ν = 0 and one uses tensor basis |i, j ≡|i, In order to use WMB method let us modify basis {|i, j } by introducing a scaling parameter κ.
One replaces μ 2 /2 and |j, m by κ 2 μ 2 /2 and |j, m, κ .This leads to a modified basis {|i, j, κ } with With this basis ( 14) is written as the matrix equation One finds μ(rad) , (μ 2 + ν 2 )∂ 2 /∂z 2 respectively.Using this scaling, ( 19) is transformed into The matrix elements for the first two terms are Matrix elements of the third term8 needs tedious but straightforward calculation9 .
and the necessary integration formula is The case a = 0 is straightforward; for a = 1, 2, one uses the integral of the type Now that we have obtained the matrix elements, our procedure goes parallel to that in subsection 2.1.Corresponding to (6) we introduce the parameter Then one can solve (21) as the standard eigenvalue problem and can obtain eigenvalues Λ n = 2/κ n .Energy levels are determined by It is found numerically that the best value of ǫ satisfies which is similar to (10).Finally we should add that we have found precise agreement between our calculations by the Sturmian basis and by the harmonic oscillator basis.10

Level repulsion
Let us first look at the nearest neighbor level spacing distribution P(s) that is the probability for the nearest neighbor level appears at the distance sΔ.H e r eΔ is the mean level spacing and after the stretch (9), Δ = 1 in this subsection.First we recollect the RMT predictions.
For γ near 1 the overlap of wave functions are negligible and P(s) is expected to be Poisson.The mean squared deviation (MSD) of level spacings is then unity.On the other hand, for large anisotropy, the wave function overlap is sizable and quantum levels repel each other.
Especially at the ergodic limit, the statistics is expected to be Wigner-Dyson(WD) statistics.
Since AKP respects time reversal invariance, one expects the limiting level statistics is WD with β = 1 (GOE).(BGS conjecture).At this limit, the RMT prediction for the MSD of level spacing is as low as 4/π − 1 = 0.273 •••.
With this theoretical expectation in mind, let us first try a coarse analysis to obtain rough idea on where the level statistics is like WD distribution using data on P(s).As the limiting MSD is 0.273, we set a condition MSD 0.28.We indicate in Fig. 1 by vertical lines the regions where this condition is met.Here we observe that, except for γ = 0.9 (the first panel), whole of the region of reliable data (the region under the mountain) remarkably satisfy the condition, while other region does not.This indicates some consistency in our analysis.But, of course, looking only at the second moment is insufficient to tell the real shape of the distribution is like-WD.Therefore, in order to step forward, we have calculated the P(s) from our data extensively.In Fig. 5, we show real P(s) distribution at γ = 0.9, 0.8, 0.7 as samples and compare with the prediction at limits (Poisson and GOE).We find that for γ 0.7 the P(s) is WD, and around γ ∼ 0.8 a small deviation from WD starts.(But still satisfies above like-WD criterion).Now the P(s) at γ = 0.9 is sizeably deviated from WD distribution.This explains why the like-WD criterion is not met at γ = 0.9.Furthermore, the deviation occurs mainly around s = 1 and remarkably the level repulsion still persists (P(s) ∝ s).From this, it seems that AKP is not a system to have the Berry-Robnik distribution (Berry & Robnik, 1984) in which the P(0) takes finite value depending on the partition of regular and chaotic orbits and rather a system that follows the stochastic approach (Hasegawa et al., 1988;Yukawa & Ishikawa, 1989).The persisting repulsion at the transitive region reminiscent to the fact that, in the Anderson transition, not only in the metallic phase but also at the mobility edge the level repulsion occurs (Fyodorov & Mirlin, 1997).Noting that AKP is not a KAM system, it would be interesting to look at the change of the behavior of P(s) near γ = 1 though it is difficult to apply proper stretch there.

Level statistics as seen by Σ 2 (L) and Δ 3 (L)
As further statistical quantities one can consider the number variance Σ 2 (L) and the spectral rigidity Δ 3 (L).These are both related to the level density-density correlation functions and suitable to the test for the RMT predictions.The number variance is defined as and the spectral rigidity 11 by Here n(L, E) is the number of (unfolded) levels within the band with L around E, N(E) is the stair case function, and ••• implies the spectral average.Because both Σ 2 (L) and Δ 3 (L) are derived from the two-point correlation functions, they are related by the Pandey relation (Pandey, 1979) This gives, for the Poisson statistics (independent levels), Δ 3 (L)=L/15 from Σ 2 (L)=L.A s amapfromΣ 2 (L) to Δ 3 (L), this gives enhanced weight for the low L side of Σ 2 (L).

Critical random matrix theories
There is an important model by García-García & Verbaarschot (2003) (we call it GV model) of the critical level statistic for the T invariant system.It is an extension of the generalized GUE (Moshe et al., 1994) to generalized GOE.This extension is successfully performed by two threads of ideas; a map to the Calogero-Sutherland (CS) model (Calogero, 1969a;b;1971;Sutherland, 1971a;b), which avoids the difficult integration over matrices in the GOE case, and the use of Kravtsov-Tsvelik (KT) conjecture for the density-density correlation function of the CS model in the low temperature limit (Kravtsov & Tsvelik, 2000).The model is defined by the joint probability distribution where the matrices S and M are both N × N and real symmetric and orthogonal respectively.This is a one-parameter model; the parameter b interpolating two statistics.At b → 0, the model becomes GOE (P(S) ∼ e − 1 2 Tr S S T ).At b → ∞,s u c hS dominate that commute with arbitrary diagonal orthogonal matrices, i.e. the Poisson ensemble.The critical statistics is obtained if b is scaled as b = h 2 N 2 for the large N limit.The joint distribution of the eigenvalues

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Theoretical Concepts of Quantum Mechanics www.intechopen.comwhere Δ(x)=∏ i<j (x i − x j ).Now, by considering the propagator of the matrix S in the imaginary time τ (Zirnbauer & Haldane, 1995), one obtains where ĤCS;λ=1 in the left-hand side is the CS hamiltonian 12 with λ = 1.Therefore, by comparing ( 33) and ( 34), one finds that the joint eigenvalue distribution of GV model is given by the diagonal matrix elements of the N-particle density matrix of CS model at an inverse temperature τ given by the identification ω/sinh ωτ = 2b, ω cosh ωτ/sinh ωτ = 2b + 1.
The KT conjecture gives the low temperature limit of connected density-density correlation function of CS model at λ = 1as where K T (x,0)=Tsin(πx)/sinh(πxT) is the kernel of CS model for λ = 2a n dT = πh/2.(García-García & Verbaarschot, 2003) suggests to replace it by its finite temperature analog (Moshe et al., 1994) and then the unfolded spectral kernel is given by13 (37) From the density-density correlation function ( 36) with ( 37) one can calculate Σ 2 (L) by and the spectral rigidity Δ 3 (L) through the Pandey relation (29).The above summarizes the work by García-García & Verbaarschot (2003).Now we are in a position to compare their predictions with the AKP data with respect to the variation of γ.We have verified, first of all, the AKP data of Σ 2 (L) and Δ 3 (L) satisfy the Pandey relation in order to guarantee that AKP levels are normal statistical set (Pandey, 1979).Now in order to test that the GV model can explain Σ 2 (L) and Δ 3 (L) coherently, we have performed a one parameter fit, at each γ,f o rt h eb e s th that explains Σ 2 (L) data and Δ 3 (L) data independently.Since the Σ 2 (L) soon shows large fluctuation for large L,wehave limited to L 5fortheΣ 2 (L) fit, and used L 20 for the Δ 3 (L) fit.The level set we used is the largest one that is obtained by the diagonalization of 19321 × 19321 (N p = 277) hamiltonian matrix (at each γ) and we used the reliable 10 4 levels starting from 2501-th.The result exhibited in Fig. 7 succinctly shows that the GV model well describes the AKP data in the range of γ from 0.75 up to 0.9.The discrepancy between the Σ 2 (L) fit and Δ 3 (L) fit is 12 The CS hamiltonian is negligible.The parameter h is related to the temperature T of the CS model as T = πh/2.We note that the correspondence between h and anisotropy γ is well described by a simple approximation h ∝ e 7.2γ .(39) In the physics of Anderson transition, a finite size scaling analysis turns out vital to scrutinize the phase transition point (García-García & Wang, 2008;Shklovskii et al., 1993).In order to compare with it, we are now trying a similar analysis using our results at various sizes.So much for the eigenvalues.Now let us turn to the wave functions.

Wave function features in anisotropic kepler problem
Here we describe three important features of AKP wave functions.Firstly, we show the systematic increase of the complexity of wave function nodal lines ({r|Ψ i (r)=0}) with the increase of the anisotropy 1 − γ.Secondly, the probability density |Ψ i (r)| 2 is investigated and it is shown that salient scars of periodic orbits are observed.Thirdly, we describe the method to evaluate Husimi functions using the basis given in subsection 2.3 and compare the Husimi distributions with the Poincaré section of the above scarring periodic orbits.

Nodal lines of AKP wave functions
The systematics of nodal lines of the eigenfunction of the laplacian operator has long history.
For instance, we can find an amazing example in (Courant&Hilbert, 1953) of a self-avoiding long nodal line of an eigenfunction of a laplacian, that propagates in the whole rectangle only by itself (Fig. 8).

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Theoretical Concepts of Quantum Mechanics www.intechopen.comIn general, the nodal line for the wave function in the non integrable case is self-avoiding.At the would be crossing, the function must be zero, and also it must be a saddle point.Both conditions can be met only coincidentally.We refer to (Gutzwiller, 1990) for nodal lines.Also we refer to a recent interesting example, the nodal line of the Maass wave form on modular surface PSL(2, Z) \ H calculated by (Hejhal & Rackner, 1992) and the increase of the complexity at higher eigenvalues.In Fig. 9 we exhibit the nodal line of AKP wave function Ψ 438 (r) at γ = 1, 0.95, 0.8, 0.2. 14One observes that even a small anisotropy, the crossings of nodal lines at the integrable limit (γ = 1) are resolved.With increasing anisotropy, nodal loops are created, and nodal lines increase their complexity.In the ergodic region (γ = 0.2), nodal lines become very complex.It seems that nodal lines in the large anisotropy region show some fractal structure.(Compare the right magnified diagram with the left one at γ = 0.2.)We are pursuing this issue introducing manifold with smoothened Coulomb singularity.The relation between the multi-fractality of wave functions (see section 5) is also under survey.

Large value of wave functions and periodic orbits
Let us try a straight forward comparison between the probability distribution of the AKP electron predicted by the wave function and periodic orbits.We have constructed and scanned |Ψ i (r)| 2 for all of the wave functions up to some 5000-th level for anisotropy γ from 0.05 to 0.98.For intermediate γ (0.85 − 0.5), the pattern varies level by level almost randomly, but we observe that there are recognizably characteristic patterns (around ten or so) that appear repeatedly.If we pick one level at random, and calculate its |Ψ(r)| 2 , the pattern is similar to one of the characteristic patterns or a combination of a few of them.On the other hand, for higher anisotropy γ < 0.5, the probability pattern becomes so complex that we cannot identify characteristic pattern.In Fig. 10, we exhibit two of characteristic patterns at γ = 0.6, |Ψ 438 | 2 and |Ψ 579 | 2 on the μ-ν plane.We have already stored periodic orbits by increasing order of the Bernoulli coding up to 8 binary digits.(For this task we benefited from a paper (Gutzwiller, 1981) which gives classification of AKP periodic orbits considering time-reversal and symmetries around axes.)In the figure selected periodic orbits that run on the μ-ν plane along the large value region of the probability distribution are also exhibited.These are the scarring orbits in the sense of (Heller, 1984;1989).It seems that the association of one periodic orbit(PO) to one probability distribution is not ad hoc, since with the change of γ, both PO and large value region change keeping the association.This observation takes the advantage of one-parameter characteristic of AKP and we are consolidating this.

Husimi function and Poincaré surface of section
Husimi function for |Ψ n is Here |CHS denotes the coherent state, a Gaussian packet centered at (q 0 , p 0 ) and width b 98 Theoretical Concepts of Quantum Mechanics www.intechopen.comFollowing (Müller & Wintgen, 1994) we obtain where |j, k;1/b 2 is given by ( 18) with κ = 1/b 2 .In Fig. 11 we exhibit the Husimi function of Ψ 438 and Ψ 579 on the μ-p μ plane.(The same wave functions with Fig. 10).We observe that the Poincaré section of periodic orbits that are associated with the large probability distributions in Fig. 10 (the scarring orbits) is now clearly sitting in the midst of the large value region of Husimi function (creating scars).It is interesting to note that the fundamental periodic orbit which creates a strong scar in Ψ 438 seems to be creating an anti-scar in Ψ 579 and the same contrast holds for other two PO's too.We finally mention that the patterns in Fig. 11 are remarkably reminiscent of the classical phase space structure presented in Fig. 1 of the seminal paper (Wintgen & Marxer, 1988).We are now extensively studying the correspondence between the quantum and classical phase space structures.

Future outlook
We discuss here two of currently pursuing problems.

Physics of quantum chaos and Anderson localization
For one thing, we contemplate to investigate the relation of quantum chaos to the Anderson localization -a very extensively studied branch of quantum physics (see for instance, Evers & Mirlin (2008)).The multi-fractality of AKP wave function was conjectured by (García-García & Verbaarschot, 2003) and AKP was considered as a candidate system to test an interesting scheme Anderson transition in quantum chaos by (García-García, 2007).
The general picture of Anderson localization is as follows (Fyodorov & Mirlin, 1997).At the metallic phase, a typical wave function is extended, the overlap of wave functions of nearby energy levels leads to the repulsion, and the statistics is WD.Approaching the Anderson transition point E c , the wave functions show up multi-fractal structure.The level statistics deviates from WD, but remarkably the repulsion still persists, which is an intrinsic feature of quantum dynamics of Anderson localization.The statistics at the transition point (the mobility edge), the critical level statistics, is just between WD and Poisson.Passing the transition point into the insulator side, the wave functions become well localized, the overlap is negligible, and the level statistics is Poisson.Now we have seen, in the quantum chaos study of AKP, a very similar phenomenon (see Table 1).In order to consolidate this correspondence, a crucial test would be the verification of the multi-fractality of wave function in AKP in the region of critical level statistics.
The multi-fractal nature of the wave function at the mobility edge is reflected in the compressibility of the energy levels as seen as the one-dimensional gas where d is the system dimension and the multi-fractal dimension D 2 necessary for the R 2 (s) statistics quantity Σ 2 (L) is defined by with p = 2.As this is a key formula relating the level statistics and the wave function structure, let us briefly follow (Chalker et al., 1996a;b) for its derivation.The compressibility χ can be calculated from the spectral form factor K(t) (the Fourier transform of the two level correlation function R(s), K(t)= ∞ −∞ dse −ist/t H R(s) with the Heisenberg time t H15 )b y taking a limit (Aronov et al., 1994;Kravtsov et al., 1994) Then there is a deep formula that gives the spectral form factor K(t) by the return probability p(t).I ti si n t r o d u c e da sa probability for the wave packet originally created in a small volume V 0 = ℓ d to remain within this volume at time t with N 0 ∼ L d /V 0 .At this point the physics of energy levels meet the physics of eigenfunctions.The return probability should behave as at the mobility edge (Chalker & Daniell, 1988;Chalker, 1990;Huckestein & Schweitzer, 1994).Now, inserting (49) into (47) and noting that the first term in the denominator is negligible compared with the second term for t ≪ t H and at L → ∞, one ultimately obtains (44).
Coming back to AKP, we have seen already that the Σ 2 (L) for large L is linearly rising in the critical statistics region.Based on this observation (and with further support of kicked particle) García-García and Verbaarschot (García-García & Verbaarschot, 2003) conjectured the multi-fractal structure of AKP wave functions in this region.We are currently working to verify quantitatively this conjecture based on (49) by our AKP data.Once the comparison of Table 1 is established, it would also give a supporting contribution to the extremely ambitious Anderson localization in quantum chaos by García-García & Wang (2008).

A non-trivial test of periodic orbit theory in AKP
Our another concern is to understand the spectral rigidity Δ 3 (L) of AKP from the periodic orbit theory following the seminal paper by (Berry, 1985).In terms of periodic orbits, the behavior of Δ 3 (L) for L ≪ L max is predicted universally by the contribution of very long classical orbits under the sum rule by (Hannay & Ozorio de Almeida, 1984). 16On the other hand, using a semi-classical sum rule, it is shown that the behavior of Δ 3 (L) for L L max is non-universally determined by the short periodic orbits.More precisely, it is of order h−(N−1) for integrable models, and log(h −1 ) for chaotic systems; the discrepancy comes from the manner of quantum interference.As we have described in the introduction, the isolated unstable orbits in AKP is symbolically coded by binary digits (Bernoulli sequences) and there is an amazing formula found by (Gutzwiller, 1980;1981) that gives a good estimate of the action for each periodic orbits concisely and enables one to evaluate the contribution of whole periodic orbits.We have already checked, using our Δ 3 (L) data, that the above L max seems to give a right value for the change of Δ 3 (L) from logarithmic rise to the asymptotic plateau.We are now trying to explain the plateau values of Δ 3 (L) data for various γ from the contribution of short periodic orbits of AKP.

Conclusion
AKP is an old working ground which produced fruitful results on quantum chaos, especially via the pioneering works by Gutzwiller (Gutzwiller, 1971;1977;1980;1981;1982;1990), and via the work by Wintgen (Wintgen & Marxer, 1988).It seems however that the focus of quantum chaos study has been shifted to elsewhere though for us it seems many things are still waiting for clarification in AKP.In this article we have extensively revisited AKP and have shed lights on its quantum features from the critical random matrix theories and from the insights from Anderson transition theories.We have in particular devoted ourselves to the quantitative investigation how the anisotropy in AKP affects systematically the quantum features of AKP.To this end, we have calculated quantum levels and wave functions from scratch.In section 2, we have recapitulated the vital WMB method (Wintgen et al., 1987) for the quantum levels.This method includes a key parameter ǫ and the effect in the eigenvalue calculation is exhibited in Fig. 1.Based on it, we have presented a prescription how to select the best ǫ for a given γ, and we have given simple rules (Fig. 3, 4).Besides the original WMB in AKP using the Sturmian basis, we have also formulated WMB calculation of AKP in terms of tensored harmonic basis.This in one hand provides us with a precise check of eigenvalues, and on the other hand, with necessary data for the Husimi function calculation.In section 3, 16 L max ∼ h−(N−1) for a system of N freedoms.

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Theoretical Concepts of Quantum Mechanics www.intechopen.comwe have investigated the energy level statistics and have shown that GV model (García-García & Verbaarschot, 2003) successfully describes the statistics in a range 0.75 γ 0.9.The finite scaling property is under survey.We have furthermore obtained a simple rule h ∝ e 7.2γ that relates the effective temperature in the CS model (equivalent to GV random matrix model) to the AKP anisotropy.In section 4 we have investigated the wave functions and Husimi functions.In both, salient quantum scar of classical unstable periodic orbits are observed.We have found that the nodal line of the wave function (at given energy) increases its complexity and seems to extend a fractal structure.In section 5 we have discussed our current projects, one on the multi-fractality of wave functions that may be deeply related to the Anderson localization, and the other on the non-trivial test of the periodic orbit theory in terms of the finite non-universal asymptote in Δ 3 (L).

Fig. 1 .
Fig.1.Stretched level curves f i at every 200 level along with the good domain on the f i − ǫ plane.N p =177, N = 7921, ℓ=even, m=0.Sky-and dark-blue mountain-like curves account for necessary conditions 1 and 2 respectively and the bulk under the curves satisfies sampling test (e.g.Fig.2).In order to make curves, diagonalization is performed for each γ and for thirty values of scaling parameter ǫ ranging from −0.3 to −0.01 with inclement 0.01.The green dashed curve connects the minimum point of f i and hits naturally the peak of the mountain.The level group with level-repulsion property is marked by red; these groups together remarkably overlap the domain for any γ (except the one γ = 0.9 > 8/9).For discussion on this, see subsection 3.1.

Fig. 2 .
Fig.2.Comparison of the level statistics P(s), Σ 2 (L), Δ 3 (L) sampled at regions (a), (b), (c).Note that graphs for the latter two statistics are properly shifted to avoid their overlap.The vertical axis is for (c).

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Anisotropic Kepler Problem and Critical Level Statistics www.intechopen.comfor large L and in the small L region we give numerical estimate.For γ = 0.6 and lower, the statistics become indistinguishable from the GOE.

Fig. 7 .
Fig. 7.The parameter h in the critical random matrix model (García-García & Verbaarschot, 2003) as determined by number variance (box with solid error bar) and by spectral rigidity (triangle with dashed error bar).

Fig. 9 .
Fig. 9. Nodal lines of the AKP wave function Ψ i (r) with i = 438 in the sector ℓ = even, m = 0 in the x − z plane.(z axis, vertical, is the heavy axis).γ = 1, 0.95, 0.8, 0.2 for (a), (b),(c), [(d), (e)] respectively.The energies are approximately equal for all, and for the integrable case (a) Ψ n p =41,ℓ=20,m=0 (r) is chosen among the degenerate levels.Two diagram are shown for γ = 0.2.The right is a magnified plot of r ≤ 0.2 part of the left one.