Supersymmetric Quantum Mechanics: two factorization schemes, and quasi-exactly solvable potentials

We present the general ideas on SuperSymmetric Quantum Mechanics (SUSY-QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its supercharges, which are defined as matrix or differential operators. We show that, although most of the SUSY partners of one-dimensional Schr\"odinger problems have already been found,there are still some unveiled aspects of the factorization procedure which may lead to richer insights of the problem involved.


I. INTRODUCTION
We present the general ideas on SuperSymmetric Quantum Mechanics (SUSY-QM) using different representations for the operators in question, which are defined by the corresponding bosonic Hamiltonian as part of SUSY Hamiltonian and its supercharges,Q − andQ + , which are defined as matrix or differential operators. We show that, although most of the SUSY partners of one-dimensional Schrödinger problems have already been found, [1] there are still some unveiled aspects of the factorization procedure which may lead to richer insights of the problem involved. In particular, we refer to the factorization of the Hamiltonian in terms of two non-mutually-adjoint operators. [2,3] In this work we try three main schemes, the first one consists on finding the eigenvalue Schrodinger equation in one dimension using the matrix representation via the appropriate factorization with ladder like operators, and finding the one parameter isospectral equation for this one. In this scheme the wave function is written as a supermultiplet. Continuining with the Schrodinger model, we extend SUSY to include two parameters factorizations, which include the SUSY factorization as particular case. As examples, we include the case of the harmonic oscillator and the Pöschl-Teller potentials. Also, we include the steps for the two-dimensional case and apply it to particular cases. The second scheme uses the differential representation in Grassmann numbers, where the wave function can be written as an n-dimensional vector or as an expansion in Grassmann variables multiplied by bosonic functions. We apply the scheme in two bosonic variables a particular cosmological model and compare the corresponding solutions found. The third scheme trias on extensions to the SUSY factorization, and to the case of quasi-exactly solvable potentials; we present a particular case which does not form part of the class of potentials found using Lie algebras.
To establish the different approaches presented here, we will briefly describe the different main formalisms applied to supersymmetric quantum mechanics, techniques that are now widely used in a rich spectrum of physical problems, cover such diverse fields as particle physics, quantum field theory, quantum gravity, quantum cosmology and statistical mechanics, to mention some of them: variables. In this formalism, a differential representation is used for the Grassmann variables. Also the supercharges for the n-dimensional case read aŝ where S is known as the super-potential function which are related to the physical potential under consideration, when the hamiltonian density is written as the Hamilton-Jacobi equation, and the algebra for the variables ψ µ andψ ν is, There are two forms where the equations in 1-D are satisfied: in the literature we find either the matrix representation or the differential operator scheme. However for more than one dimensions, there exist many applications to cosmological models, where the differential representation for the Grassmann variables is widely applied [14][15][16][17][18]. There are few works in more dimensions in the first scheme [19], we present in this work the main ideas to built the 2D case, where the supercharges operators become 4 × 4 matrices.

II. FACTORIZATION METHOD IN 1-DIMENSION: MATRIX APPROACH
We begin by introducing the main ideas for the 1-Dimensional quantum harmonic oscillator . The corresponding hamiltonian is written in operator form aŝ whereq is the generalized coordinate, andp is the associated momentum, the canonical commutation relation between this quantities being [q,p] = i . We introduce two new operators, known as the creation and annihilation operatorsâ + ,â − respectively, defined aŝ This hamiltonian can be written in terms of the anti-commutation relation between these operators aŝ the symmetric nature ofĤ B under the interchange ofâ − andâ + suggests that these operators satisfy Bose-Einstein statistics, and it is therefore called bosonic. Now, we build the operatorsb − andb + that obey similar rules to operatorsâ − ,â + changing [ , ] { , }, that is and in analogy to (5), we define the corresponding new hamiltonian aŝ The antisymmetric nature ofĤ F under the interchange ofb − andb + suggests that these operators satisfy the Fermi-Dirac statistics, and it is called fermionic. These operatorsb − andb + admit a matrix representations in terms of Pauli matrices, that satisfy all rules defined above, that isb Now, consider both hamiltonians as a composite system, that is, we consider the superposition of two oscillators, one being bosonic and one fermionic, with energy When we demand that both frequencies are the same, ω B = ω F = ω, we introduce a new symmetry, called supersymmetry (SUSY), we can see that the simultaneous creation of a quantum fermion (n F → n F + 1), causes the destruction of quantum boson (n B → n B − 1) and viceversa, in the sense that the total energy is unaltered. The ground energy state is exact and no degenerate. The degeneration appears from n=1, where it is double degenerate.
In this way, we have the super-hamiltonianĤ susy , written aŝ where I is a 2 × 2 unit matrix, and where the two components ofĤ susy in (10) can be written independently aŝ From equations (18) and (19), we can see thatĤ + andĤ − are the same representation of one hamiltonian with a constant shifting ω in the energy spectrum. The question is, what are the generators for this SUSY hamiltonian? The answer is, considering that the degeneration is the result of the simultaneous destruction (creation) of quantum boson and the creation (destruction) of quantum fermion, that the corresponding generators for this symmetry must be written asâ −b+ (orâ +b− ). therefore we introduce the following generators, called superchargesQ − andQ + defined aŝ implying thatĤ and satisfying the following relations We can generalize this procedure for a certain function W(q), and at this point we can define two new operatorsÂ − andÂ + with a property similar to (4), In order to obtain the general solutions, we can use an arbitrary potential in equation (3), that iŝ the hamiltoniansĤ + andĤ − determine two new potentials, where the potential term V + (q) is related to the superpotential function W(q) via the Ricatti equation (modulo constant , which is related to some energy eigenvalue) and V − = 1 2 W 2 + dW dq = V + + dW dq , with the same spectrum, except for the ground state, which is related to the energy potential V + .
In a general way, let us now find the general form of the function W. The quantum equation (17) applied to stationary wave function u i becomes where E i are the energy eigenvalues. Considering the transformation W(q) = − dln[ui(q)] dq and introducing it into (18), we have that then, this equation is the same as the original one, eq.(21), that is, W is related to a initial solution of the bosonic hamiltonian. What happens to the iso-potential V − (q) = 1 2 W 2 + dW dq ? Considering that the question is, what isŴ if we know the function W? Finding it we can build a family of potentialsV − depending on a free parameter λ, the supersymmetric parameter that, to some extent, plays the role of internal time. Following the procedureŴ = W + 1 y(q) , where the function y(q) satisfy the linear differential equation dy dq − 2Wy = 1, the solution implies The family of potentialsV + can be built now aŝ is the isospectral solution of the Schrödinger like equation for the new family potential (23), with the condition g(λ) = λ(λ + 1), which in the limit This λ parameter is included not for factorization reasons; in particular, in quantum cosmology the wave functions are still nonnormalizable, and λ is used as a decoherence parameter embodying a sort of quantum cosmological dissipation (or damping) distance.
A. Two dimensional case.
We use Witten's idea [20], to find the supersymmetric supercharges operators Q − and Q + that generate the superHamiltonian H susy . Using equations (13), (14) and (15), we can generalize the one-dimensional factorization scheme. We define the two dimensional Hamiltonian aŝ where the Schrödinger like equation can be obtained as the bosonic sector of this super-Hamiltonian in the superspace, i.e, when all fermionic fields are set equal to zero (classical limit).
In two dimensions the supercharges are defined by the tensorial products where σ ± are the same as in (8). From equations (26) we have that the supercharges are 4 × 4 matriceŝ where the super-Hamiltonian, (14), can be written as where and V(x, y) = W(x) + Z(y).
The Ricatti equation (20) is written in 2D as and, using separation variables, we get In general, we find that each potential V +i satisfy and we can find the iso-potential as W = − 1 u1 du1 dx , when u 1 is known. Following the same steps as in the 1D case, we find that the solutions (22) are the same in this case. So, the general solution forŴ isŴ = W + 1 y(x) , with y = u −2 where W p = − 1 u1 du1 dx and I 1 = u 2 1 dx.
In the same manner, we have that with Z p = − 1 u2 su2 dy and I 2 = u 2 2 dy. On the other hand, using the Ricatti equation, we can build a generalization for the isopotential, using the new potentialŴ, asV For the other coordinate, we havê The general solutions forû i depends on the initial solutions to the original Schrödinger equations in the variables (x,y), that is, where the variables C i (λ i ) have the same properties that g(λ) obtained in the 1D case.

B. Application to cosmological Taub model
The Wheeler-DeWitt equation for the cosmological Taub model is given by where V(β) = 1 3 e −8β − 4e −2β . This equations can be separated using x 1 = 4α − 8β and x 2 = 4α − 2β, rendering where the parameter ω is the separation constant. These equations possess the solutions where K (or I) is the modified Bessel function of imaginary order, and the functions L is define as Using equations (38) and (39) we obtain the isopotential for this model (44) Using (40) we can obtain general solutions for the functions f 1 and f 2 in the following waŷ

III. DIFFERENTIAL APPROACH: GRASSMANN VARIABLES
The supersymmetric scheme has the particularity of being very restrictive, because there are many constraint equations applied to the wave function. So, in this work and in others, we found that there exist a tendency for supersymmetric vacua to remain close to their semiclassical limits, because the exact solutions found are also the lowest-order WKB like approximations, and do not correspond to the full quantum solutions found previously for particular models. [14][15][16][17][18] Mantaining the structure of the equations (13), (14), (15) and (16), taking the differential representation for the fermionic operatorb ↔ ψ µ for convenience in the calculations, and changing the function W → ∂S ∂q µ , the supercharges for the n-dimensional case read aŝ where S is known as the super-potential functions which are related to the physical potential under consideration, when the hamiltonian density is written as the Hamilton-Jacobi equation, and the following algebra for the variables ψ µ andψ ν , (similar to equation (6)) these rules are satisfied when we use a differential representation for these ψ µ ,ψ ν variables in terms of the Grassmann numbers, as where η µν is a diagonal constant matrix, its dimensions depending on the independent bosonic variables that appear in the bosonic hamiltonian. Now the superhamiltonian is written as where H 0 = + U(q µ ) is the quantum version of the classical bosonic hamiltonian, is the d'Alambertian in three dimension when we have three bosonic independent coordinates, and U(q µ ) is the potential energy in consideration.
The superspace for three dimensional model becomes (q 1 , q 2 , q 3 , θ 0 , θ 1 , θ 2 ), where the variables θ i are the coordinate in the fermionic space, as the Grassmann numbers, which have the property of θ i θ j = −θ j θ i , and the wavefunction has the representation where the indices µ, ν, λ values are 0,1 and 2, and A ± , B ν and C λ are bosonic functions which depend on the bosonic coordinates q µ and not on the Grassmann numbers. Here, the wavefunction representation structure is set in terms of 2 n components, for n independent bosonic coordinates, with half of the terms coming from the bosonic (fermionic) contribution into the wavefunction. It is well known that the physical states are determined by the applications of the superchargesQ − andQ + on the wavefunctions, that isQ where we use the usual representation for the momentum P µ = −i ∂ ∂q µ . Considering the 2D case, the last second equation gives from (54) and (55) we obtain the relation ∂A+ ∂q µ − A + ∂S ∂q µ = 0 with the solution A + = a + e S . On the other hand, the first equation in (53) gives the free term equation is written as η µν (∂ µ B ν + B ν ∂ µ S) = 0, and taking the ansatz B µ = e −S ∂ ν f + (q µ ), the equation (56) is fulfilled, so we obtain for the free term, with the solution to f + = h(q 1 − q 2 ), with h an arbitrary function depending of its argument. However, this function f must depend on the potential under consideration. Also, equations (57) and (58) are written as In this way, all functions entering the wavefunction are

A. The unnormalized probability density
To obtain the wavefunction probability density |Ψ| 2 in this supersymmetric fashion, we need first to integrate over the Grassmann variables θ i . This procedure is well known, [21] and here we present the main ideas. Let Ψ 1 and Ψ 2 be two functions that depend on Grassmann numbers, the product < Ψ 1 , Ψ 2 > is defined as and the integral over the Grassmann numbers is θ * i θ i · · · θ m θ * m dθ * m dθ m · · · dθ * i dθ i = 1. In 2D, the main contributions to the term e − i θ * i θi come from e − i θ * i θi = e i θiθ * i = 1 + θ 0 θ * 0 + θ 1 θ * 1 + θ 0 θ * 0 θ 1 θ * 1 and using that θdθ = 1, and dθ = 0, which act as a filter, we obtain that By demanding that |Ψ| 2 does not diverge when |q 0 |, |q 1 | → ∞, only the contribution with the exponential e −2S will remain.

IV. BEYOND SUSY FACTORIZATION
Although most of the SUSY partners of 1D Schrödinger problems have been found, [1] there are still some unveiled aspects of the factorization procedure. We have shown this for the simple harmonic oscillator in previous works, [2,3] and will procede here in the same way for the problem of the modified Pöschl-Teller potential. The factorization operators depend on two supersymmetric type parameters, which when the operator product is inverted allow us to define a new SL operator, which includes the original QM problem. The Hamiltonian of a particle in a modified Pöschl-Teller potential is [1,22] where α > 0, and the integer m is greater than 0. To shorten the algebraic equations we shall set 2 2µ = 1. The eigenvalue problem may be solved using the Infeld & Hull's (IH) factorizations, [23] where the IH raising/lowering operators are given by where k(x, m) = αm tanh αx; also m = α 2 m 2 , and n is the eigenvalue index, Beginning with the zeroth order eigenfunctions The eigenfunctions can be found by successive applications of the raising operator, which only increases the value of the upper index. That is, we repeatedly apply the creation operator A − s+1 ψ s = ψ s+1 . Note that from (63), A − m A + m and A + m A − m give different Hamiltonian operators.

A. Two parameter factorization of the Pöschl-Teller Hamiltonian
Following our previous work, [2,3] we define two non-mutually adjoint first order operators, where β m and η m are functions of x, and we require that B m+1 B * m+1 = H m+1 + m+1 . Then β m+1 and η m+1 are the solutions of By multiplying the first equation by β/η and adding, we have that This Ricatti equation was found in [24], it has the solution β/η = D tanh αx, with = D 2 , and two possible values for D, D = α(m + 1) , −αm. If we simply set η m → 1, we recover the factorization (63a). The constant is usually related to the lowest energy eigenvalue, but here the two different values come from the index asymmetry in the factorizations (63). Following Ref. [24], we solve for D = α(m + 1).

C. Regions in the two-parameter space
We may recover the original QM problem when γ 1 = γ 2 = 0, the origin of the two-parameter space. Moreover, the SUSY partner of the PT problem arises when one sets γ 2 = 0, moving along the horizontal axis. In this case, L becomes where λ = m + 1, with S 1 (αx) = γ1 sech 2λ (αx) 1+γ1 x 0 sech 2λ αy dy , and ω(x) = 1. These in turn define a SUSY PT problem where the partner SUSY potentials are given by The zero-order eigenfunction is defined by B − φ 0 = 0, that is

QUASI-EXACTLY SOLVABLE POTENTIALS
In exactly solvable problems the whole spectrum is found analytically, but the vast majority of problems have to be solved numerically. A new possibility arised with the class of QES potentials, where a subset of the spectrum may be found analytically. [25][26][27] QES potentials have been studied using the Lie algebraic method [25]: Manning, [28] Razavy [29], and Ushveridze [30] potentials belong to this class (see also [31]). Theses are double well potentials, which received much attention due to their applications in theoretical and experimental problems. Furthermore, hyperbolic type potentials are found in many physical applications, like the Rosen-Morse potential, [32] Dirac type hyperbolic potentials, [33] bidimensional quantum dot, [34] Scarf type entangled states, [35] etc. QES potentials classification have been given by Turbiner, [25] and Ushveridze. [30] Here we show that the Lie algebraic procedure may impose strict restrictions on the solutions: we shall construct here analytical solutions for the Razavy type potential V (x) = V 0 sinh 4 (x) − k sinh 2 (x) based on the polynomial solutions of the related Confluent Heun Equation (CHE) [36], and show that in that case the energy eigenvalues diverge when k → −1, a feature solely of the procedure. We shall also show that other QES potentials may be found that do not belong to any of the potentials found using the Lie algebraic method.
Note that in this case (E 1 − E 0 )/E 0 = 0.0052, and it is not possible to distinguish these eigenvalue's lines from each other in Fig.(1) for antisymmetric eigenvalues, implying quasi-degenerate eigenstates. A similar effect is seen in the symmetric case.
A. The case with k = −1 As was seen in Section VI, the ground state energy diverges as 1/(1 + k) as k → −1, and this also happens to all higher order even eigenvalues (see eq.(93)). This is a strange behaviour, since it is clear that the potential function has a rather simple functional form for any value of k: a single or double well with infinite barriers. We can see that this is only a characteristic due to the analytical solution procedure, coming from the fact that the potential strength V 0 is also divergent when k → −1.

B. Unclassified QES potentials
Finally, we would like to emphasize that there should be other potential functions which may not be classified form the Lie algebraic methood. [25] Indeed, let us consider Schrödinger's problem with the potential function For this problem, the ground state eigenfunction and eigenvalue are given by ψ = ψ 0 e −α cosh(x) cosh(x) , E = α 2 − 1 2 (96) while this particular problem does not belong to the class of potentials found using the Lie algebraic method. Similar potentials may be found which do not belong to that class, leaving space for further developments.