Open access peer-reviewed chapter - ONLINE FIRST

Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable Potentials

By José Socorro García Díaz, Marco A. Reyes, Carlos Villaseñor Mora and Edgar Condori Pozo

Submitted: August 2nd 2018Reviewed: October 25th 2018Published: March 15th 2019

DOI: 10.5772/intechopen.82254

Downloaded: 89

Abstract

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 Schrödinger 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.

Keywords

  • supersymmetric quantum mechanics
  • quasi-exactly solvable potentials

1. 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, Q̂and Q+, 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 Schrödinger 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. Continuing with the Schrödinger model, we extend SUSY to include two-parameter 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 tries 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, covering such diverse fields as particle physics, quantum field theory, quantum gravity, quantum cosmology, and statistical mechanics, to mention some of them:

  • In one dimension, SUSY-QM may be considered an equivalent formulation of the Darboux transformation method, which is well-known in mathematics from the original paper of Darboux [4], the book by Ince [5], and the book by Matweev and Salle [6], where the method is widely used in the context of the soliton theory. An essential ingredient of the method is the particular choice of a transformation operator in the form of a differential operator which intertwines two Hamiltonian and relates their eigenfunctions. When this approach is applied to quantum theory, it allows to generate a huge family of exactly solvable local potential starting with a given exactly solvable local potential [7]. This technique is also known in the literature as isospectral formalism [7, 8, 9, 10].

  • Those defined by means of the use of supersymmetry as a square root [11, 12, 13, 14], in which the Grassmann variables are auxiliary variables and are not identified as the supersymmetric partners of the bosonic variables. In this formalism, a differential representation is used for the Grassmann variables. Also the supercharges for the n-dimensional case read as

Q̂=ψμqμ+∂Sqμ,Q̂+=ψ¯νqν∂Sqν,E1

where Sis known as the superpotential 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 ψμand ψ¯νis

ψμψ¯ν=ημν,ψμψν=0,ψ¯μψ¯ν=0.E2

There are two forms where the equations in 1D 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 build the 2D case, where the supercharge operators become 4×4matrices.

2. Factorization method in one dimension: matrix approach

We begin by introducing the main ideas for the one-dimensional quantum harmonic oscillator. The corresponding Hamiltonian is written in operator form as

ĤB=12p̂2+12ωB2q̂2E3

where q̂is the generalized coordinate and p̂is the associated momentum, the canonical commutation relation between this quantities being q̂p̂=i. We introduce two new operators, known as the creation and annihilation operators â+,â, respectively, defined as

â=12ωBp̂iωBq̂,â+=12ωBp̂+iωBq̂.E4

This Hamiltonian can be written in terms of the anti-commutation relation between these operators as

ĤB=ωB2â+â.E5

The symmetric nature of ĤBunder the interchange of âand â+suggests that these operators satisfy Bose-Einstein statistics, and it is therefore called bosonic.

Now, we build the operators b̂and b̂+that obey similar rules to operators â,â+changing , that is,

b̂b̂+=1;b̂b̂=b̂+b̂+=0,E6

and in analogy to (5), we define the corresponding new Hamiltonian as

ĤF=ωF2b̂+b̂.E7

The antisymmetric nature of ĤFunder the interchange of b̂and b̂+suggests that these operators satisfy the Fermi-Dirac statistics, and it is called fermionic.

These operators b̂and b̂+admit a matrix representations in terms of Pauli matrices that satisfy all rules defined above, that is,

b̂=σ,b̂+=σ+,σ±=12σ1±iσ2E8

with σ+σ=σ3, σ=0010,σ+=0100,σ1=0110,σ2=0ii0,σ3=1001.

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 ET=EB+EF

ET=ωBnB+12+ωFnF12=ωBnB+ωFnF+12ωBωF.E9

When we demand that both frequencies are the same, ωB=ωF=ω, we introduce a new symmetry, called supersymmetry (SUSY); we can see that the simultaneous creation of a quantum fermion nFnF+1causes the destruction of quantum boson nBnB1and vice versa, 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-Hamiltonian Ĥsusy, written as

Ĥsusy=ω2â+â+ω2b̂+b̂=ω2â+âI+ω2σ3=ωââ+00â+â=Ĥ00Ĥ+,E10

where I is a 2×2unit matrix and where the two components of Ĥsusyin (10) can be written independently as

Ĥ+=12p̂2+12ω2q2ωωâ+âE11
Ĥ=12p̂2+12ω2q2+ωωââ+.E12

From Eqs. (18) and (19), we can see that Ĥ+and Ĥare the same representation of one Hamiltonian with a constant shifting ωin 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, the corresponding generators for this symmetry must be written as âb̂+(or â+b̂). Therefore we introduce the following generators, called supercharges Q̂and Q̂+, defined as

Q̂=2ωâb̂+=2ω0â00,Q̂+=2ωâ+b̂=2ω00â+0,E13

implying that

Ĥsusy=12Q̂+Q̂E14

and satisfying the following relations

Q̂Q̂=Q̂+Q̂+=0;Q̂Ĥsusy=Q̂+Ĥsusy=0.E15

We can generalize this procedure for a certain function W(q), and at this point, we can define two new operators Âand Â+with a property similar to (4),

Â=12ωp̂iωWq,Â+=12ωp̂+iωWq.E16

In order to obtain the general solutions, we can use an arbitrary potential in Eq. (3), that is,

ĤB=12p̂2+Vq.E17

The Hamiltonians Ĥ+and Ĥdetermine two new potentials,

Ĥ+=12p̂2+V+=12p̂2+12W2dWdqE18
Ĥ=12p̂2+V=12p̂2+12W2+dWdq,E19

where the potential term V+(q) is related to the superpotential function W(q) via the Riccati equation

V+=12W2dWdq,E20

(modulo constant ϵ, which is related to some energy eigenvalue) and V=12W2+dWdq=V++dWdq, 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 uibecomes

12d2uidq2+Vqui=Eiui,E21

where Eiare the energy eigenvalues. Considering the transformation Wq=dlnuiqdqand introducing it into (18), we have that

VqEi=12W2dWdq=12uiduidq2duidq2uid2uidq22ui2=12uid2uidq2.

Then, this equation is the same as the original one, Eq. (21), that is, W is related to an initial solution of the bosonic Hamiltonian. What happens to the isopotential Vq=12W2+dWdq? Considering that

2V=W2+dWdqŴ2+dŴdq=2V̂,

the question is, what is Ŵif we know the function W? Finding it we can build a family of potentials V̂depending on a free parameter λ, the supersymmetric parameter that, to some extent, plays the role of internal time. Following the procedure Ŵ=W+1yq, where the function y(q) satisfy the linear differential equation dydq2Wy=1, the solution implies

yq=λ+ui2dqui2,Ŵ=W+ui2λ+ui2dq.E22

The family of potentials V̂+can be built now as

V̂+Ei=12Ŵ2dŴdq=V+dŴdq.E23

Finally

û=gλuiλ+ui2dqE24

is the isospectral solution of the Schrödinger-like equation for the new family potential (23), with the condition gλ=λλ+1, which in the limit

λ±,gλ=λ,ûiui.

This λparameter is included not for factorization reasons; in particular, in quantum cosmology the wave functions are still nonnormalizable, and λis used as a decoherence parameter embodying a sort of quantum cosmological dissipation (or damping) distance.

2.1 Two-dimensional case

We use Witten’s idea [20] to find the supersymmetric supercharge operators Qand Q+that generate the super-Hamiltonian Hsusy. Using Eqs. (13)(15), we can generalize the one-dimensional factorization scheme. We define the two-dimensional Hamiltonian as

ĤBxy=12p̂x2+12p̂y2+Vx+Vy,E25

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

Q=2dσ+Q+=2d+σE26

with

d=a00bd+=a+00b+,E27

where σ±are the same as in (8). From Eq. (26) we have that the supercharges are 4×4matrices

Q̂+=200000000a+0000b+00Q̂=200a0000b00000000E28

where the super-Hamiltonian, (14), can be written as

Hsusy=aa+0000bb+0000a+a0000b+b=H1x0000H1y0000H+2x0000H+2y,E29

where

a=12ddx+Wx,a+=12ddx+WxE30
b=12ddx+Zy,b+=12ddx+ZyE31

and Vxy=Wx+Zy.

The Riccati equation (20) is written in 2D as

V+xy=V+1x+V+2y=12W2dWdx+12Z2dZdy,E32

and, using separation variables, we get

V1x12W2xdWdx=C0E33
V2y12Z2ydZdy=C0.E34

In general, we find that each potential V+isatisfies

12d2dx2uix+V+iuix=Eiuix,i=1,2,E35

and we can find the isopotential as W=1u1du1dx, when u1is 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 for Ŵis Ŵ=W+1yx, with y=u12xE1+u12xdx. The general solution for the superpotential Ŵxis

Ŵ=1u1du1dx+u12λ1+u12dx=Wp+ddxLnλ1+I1E36

where Wp=1u1du1dxand I1=u12dx.

In the same manner, we have that

Ẑ=1u2du2dy+u22λ2+u22dy=Zp+ddyLnλ2+I2E37

with Zp=1u2su2dyand I2=u22dy.

On the other hand, using the Riccati equation, we can build a generalization for the isopotential, using the new potential Ŵ, as

V̂+1xλ1=12Ŵ2Ŵ=V+x2u1du1dxλ1+I1+u14λ1+I12.E38

For the other coordinate, we have

V̂+2yλ2=12Ẑ2dẐdy=V+y2u2du2dyλ2+I2+u24λ2+I22.E39

The general solutions for ûidepend on the initial solutions to the original Schrödinger equations in the variables (x,y), that is, u1=u1x, u2=u2y, being

û1xλ1=C1λ1u1λ1+I1,û2yλ2=C2λ2u2λ2+I2,E40

where the variables Ciλihave the same properties that gλobtained in the 1D case.

2.2 Application to cosmological Taub model

The Wheeler-DeWitt equation for the cosmological Taub model is given by

2Ψα22Ψβ2+e4αVβΨ=0E41

where Vβ=13e8β4e2β. These equations can be separated using x1=4α8βand x2=4α2β, rendering

2f1x1x12+1144ex1f1x1=ω24f1x1,2f2x2x22+19ex2f2x2=ω2f2x2,E42

where the parameter ωis the separation constant. These equations possess the solutions

f1=Kiω16ex12,f2=L223ex22+K223ex22E43

where K (or I) is the modified Bessel function of imaginary order and the function L is defined as

L2=πi2sinh2ωπI2+I2.

Using Eqs. (38) and (39), we obtain the isopotential for this model

V̂x1=V+x12KKλ1+I1+K4λ1+I12,V̂x2=V+x22L2+K2L2+K2λ2+I2+L2+K24λ2I22.E44

Using Eq. (40) we can obtain general solutions for the functions f1and f2in the following way

f̂1=C1K16ex12λ1+I1,f̂2=C2L223ex22+K223ex22λ2+I2.E45

3. 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].

Maintaining the structure of Eqs. (13)(16), taking the differential representation for the fermionic operator b̂ψμfor convenience in the calculations, and changing the function W∂Sqμ, the supercharges for the n-dimensional case read as

Q̂=ψμPμ+i∂Sqμ,Q̂+=ψ¯νPνi∂Sqν,E46

where Sis known as the superpotential 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 ψμand ψ¯ν(similar to Eq. (6))

ψμψ¯ν=ημν,ψμψν=0,ψ¯μψ¯ν=0.E47

These rules are satisfied when we use a differential representation for these ψμ,ψ¯νvariables in terms of the Grassmann numbers, as

ψμ=ημνθν,ψ¯ν=θν,E48

where ημνis a diagonal constant matrix, its dimensions depending on the independent bosonic variables that appear in the bosonic Hamiltonian. Now the super-Hamiltonian is written as

HS=12Q̂+Q̂=H0+22Sqμqνψ¯μψν,E49

where H0=+Uqμis the quantum version of the classical bosonic Hamiltonian, is the d’Alembertian in three dimension when we have three bosonic independent coordinates, and Uqμis the potential energy in consideration.

The superspace for three-dimensional model becomes q1q2q3θ0θ1θ2, where the variables θiare the coordinate in the fermionic space, as the Grassmann numbers, which have the property of θiθj=θjθi, and the wave function has the representation

Ψ=A++B0θ0,1dimensionE50
Ψ=A++B0θ0+B1θ1+Aθ0θ1,2dimensionsE51
Ψ=A++Bνθν+12ϵμνλCλθμθν+Aθ0θ1θ2,3dimensionsE52

where the indices μ,ν,λvalues are 0, 1, and 2 and A±,Bνand Cλare bosonic functions which depend on the bosonic coordinates qμand not on the Grassmann numbers. Here, the wave function representation structure is set in terms of 2ncomponents, for nindependent bosonic coordinates, with half of the terms coming from the bosonic (fermionic) contribution into the wave function.

It is well-known that the physical states are determined by the applications of the supercharges Q̂and Q̂+on the wave functions, that is,

Q̂Ψ=0,Q̂+Ψ=0,E53

where we use the usual representation for the momentum Pμ=iℏqμ. Considering the 2D case, the last second equation gives

θ0:A+q0A+Sq0=0,E54
θ1:A+q1A+Sq1=0,E55
θ0θ1:B1q0B1Sq0B0q1B0Sq1=0.E56

From (54)(55), we obtain the relation A+qμA+Sqμ=0with the solution A+=a+eS.

On the other hand, the first equation in (53) gives

θ0:Aq1+ASq1=0,E57
θ1:Aq0+ASq0=0,E58
free term:B0q0+B0Sq0+B1q1+B1Sq1=0.E59

The free term equation is written as ημνμBν+BνμS=0, and taking the ansatz Bμ=eSνf+qμ,Eq. (56) is fulfilled, so we obtain for the free term,

f++2ημνμSνf+=0,E60

with the solution to f+=hq1q2, with h an arbitrary function depending of its argument. However, this function f must depend on the potential under consideration.

Also, Eqs. (57) and (58) are written as

Aqμ+A∂Sqμ=0,1AAqμ=∂SqμLnAqμ=∂SqμE61

whose solution is A=aeS. In this way, all functions entering the wave function are

A±=a±e±S,B0=eS0f+,B1=eS1f+.

3.1 The unnormalized probability density

To obtain the wave function probability density Ψ2in this supersymmetric fashion, we need first to integrate over the Grassmann variables θi. This procedure is well-known [21], and here we present the main ideas. Let Ψ1and Ψ2be two functions that depend on Grassmann numbers; the product <Ψ1,Ψ2>is defined as

<Ψ1,Ψ2>=Ψ1θΨ2θeiθiθiΠidθidθi,Cθiθr=θrθiC,

and the integral over the Grassmann numbers is θiθiθmθmdθmdθmdθidθi=1.

In 2D, the main contributions to the term eiθiθicome from

eiθiθi=eiθiθi=1+θ0θ0+θ1θ1+θ0θ0θ1θ1,

and using that θdθ=1, and =0, which act as a filter, we obtain that

Ψ2=A+A++B0B0+B1B1+AA.

By demanding that Ψ2does not diverge when q0,q1, only the contribution with the exponential e2Swill remain.

4. 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 proceed 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]

Hm+1Ψ=22μd2dx2α2mm+1cosh2αxΨ=EΨ,E62

where α>0and the integer mis greater than 0. To shorten the algebraic equations, we shall set 22μ=1.

The eigenvalue problem may be solved using the Infeld and Hull’s (IH) factorizations [23],

Am+1+Am+1ψmnm=Hm+1+ϵm+1ψmnm,E63
AmAm+ψmnm=Hm+1+ϵmψmnm,E64

where the IH raising/lowering operators are given by

Am=kxmddxE65

and where kxm=αmtanhαx; also ϵm=α2m2, and nis the eigenvalue index,

Ψn=ψmnm,En=ϵmn=α2mn2,n=0,1,2<m.E66

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,

ψx=αΓ+12πΓcoshαx.E67

We repeatedly apply the creation operator As+1ψs=ψs+1. Note that from (63) and (64), AmAm+and Am+Amgive different Hamiltonian operators.

4.1 Two-parameter factorization of the Pöschl-Teller Hamiltonian

Following our previous work [2, 3], we define two non-mutually adjoint first-order operators,

Bm=ηm1ddx+βm,Bm=ηmddx+βm,E68

where βmand ηmare functions of x, and we require that Bm+1Bm+1=Hm+1+ϵm+1. Then βm+1and ηm+1are the solutions of

ηη+βηβη=0,βη+β2=α2mm+1cosh2αx+ϵ.E69

By multiplying the first equation by β/ηand adding, we have that

βm+1ηm+1+βm+1ηm+12=α2mm+1cosh2αx+ϵm+1.E70

This Riccati equation was found in [24]; it has the solution β/η=Dtanhαx, with ϵ=D2, and two possible values for D, D=αm+1,αm. If we simply set ηm1, we recover the factorization (63).

The constant ϵis usually related to the lowest energy eigenvalue, but here the two different values come from the index asymmetry in the factorizations (63) and (64). Following Ref. [24], we solve for D=αm+1.

The general solution to the pair of coupled equations (69) is

ηm+1x=1+γ2sech2m+1αx1+γ10xsech2m+1αydy21/2,E71

and

βm+1x=αm+1tanhαx+γ1sech2m+1αx1+γ10xsech2m+1αydy×ηm+1x,E72

where γ1has to satisfy γ1<2αΓm+3/2/πΓm+1. The corresponding condition on γ2involves transcendental functions, but one may use γ2>1+γ12to determine the γ1γ2parameter space. When γ1=γ2=0we recover the original IH raising/lowering operators.

4.2 Reversing the operator product: new Sturm-Liouville operator

Now we invert the first-order operators’ product, keeping in mind Eq. (64),

BmBm=d2dx2+2ηmηmddx+V0+ϵmηmβmβmηm.E73

Then we can define a new Sturm-Liouville (SL) eigenvalue problem LΦn+ωxEnΦn=0, where

L=ddxηm2ddx+ϵmβm21+ηm2α2mm+1sech2αxE74
Φn=ϕmnmBmψmnm1,E75

with the weight function ωx=ηm2x.

This new SL operator is isospectral to the original PT problem. The zeroth-order eigenfunction is easily found by solving Bϕ0=ddx+βmηmΦ0=0which gives

Φ0=ηmx×sechm+1αx1+γ10xsech2m+1αydy.E76

4.3 Regions in the two-parameter space

We may recover the original QM problem when γ1=γ2=0, the origin of the two-parameter space. Moreover, the SUSY partner of the PT problem arises when one sets γ2=0, moving along the horizontal axis. In this case, Lbecomes

L=d2dx2+α2λλ+1sech2αx2S12αx4αλtanhαxS1αxE77

where λ=m+1, with S1αx=γ1sech2λαx1+γ10xsech2λαydy, and ωx=1. These in turn define a SUSY PT problem

d2dx2+V˜xΦn=EnΦnxE78

where the partner SUSY potentials are given by

V˜=α2λλ+1sech2αx+2S12αx+4αλtanhαxS1αx.E79

The zeroth-order eigenfunction is defined by Bϕ0=0, that is,

ϕ0=sechλαx1+γ10xsech2λαydy.E80

5. 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 arises 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]). These 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 has 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 Vx=V0sinh4xksinh2xbased on the polynomial solutions of the related confluent Heun equation (CHE) and show that in that case the energy eigenvalues diverge when k1, 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.

5.1 A Razavy-type QES potential

Let us consider Schrödinger’s problem for the Razavy-type potential Vx=V0sinh4xksinh2x,

22μd2ψxdx2+V0sinh4λxksinh2λxψx=Eψx.E81

For simplicity, we set μ==λ=1[35, 36].

Here the potential function is the hyperbolic Razavy potential Vx=12ζcosh2xM2, with V0=2ζ2, where Menergy levels are found if Mis a positive integer [29]. It may also be viewed as the Ushveridze potential Vx=2ξ2sinh4x+2ξξ2γ+δ2sinh2x+2δ14δ34csch2x2γ14γ34sech2x, when γ=14and δ=34, or vice versa [30], which is QES if =0,1,2,(with δ14). El-Jaick et al. showed that it is also QES if =half-integer and γ,δ=14,34[37].

In the case of the Razavy potential, the solutions obtained by Finkel et al. are

ψσηxERsinhx121σηcoshx121σ+ηeζ2cosh2xj=0nP̂jσηER2j+ησ+12!cosh2jxE82

with the parameters ση=±10or 0±1, the energy eigenvalues being the roots of the polynomials Pj+1σηER, satisfying the three-term recursive relations

P̂j+1ση=ERbjP̂jσηERajP̂j1σηER,j0E83

with ER=2E, and

aj=16ζj2jσ+ηjn1bj=4jj+1σ+2ζ+2n+12nσ+3+ζζ2η+4n.E84

5.2 Symmetric solutions for Vx=V0sinh4x

To find the even solutions to Eq. (81) with k=0, let us set βx=cosh2x, to get

ββ1d2ψdβ2+β12+142E2V0β2+4V0β2V0=0E85

and to ensure that ψxvanishes as x±, let ψx=eα2βfβ. Previous works may not include square-integrable solutions to the Razavy potential [38, 39, 40]. By requiring α2=2V0, we obtain [41]

ββ1d2fdβ2+αββ1+β12df+α2β4αβ2+α4+E2α24f=0.E86

We shall look for rank Npolynomial solutions: fβ=f0for N=0, or fβ=f0i=1Nββifor N>0, the βibeing the roots of the resulting polynomial in Eq. (86). Sometimes the N=0solution is not even considered [35].

The highest power of βin Eq. (86) fix αto α=4N+2. The energy eigenvalues and the roots satisfy

E=12α2+α4i=1Nβi14N4N2E87
ijN2βiβj+αβi2+α+1βi12βi2βi=0,i=1,2,,n.E88

V0is found to depend on the order of the polynomial, V0=22N+12for even solutions, and solutions with different Ncannot be scaled one into the other due to the sinh4xdependence of the potential function. The highest solution order is n=2N, and we use subindexes Nnto label eigenvalues/eigenfunctions.

For N=0, fβ=1, we get V0=2, E0,0=1, and the (unnormalized) ground-state eigenfunction ψ0,0x=ecosh2x. For N=2, fβ=f0ββ1ββ2, equating to zero the coefficients of the polynomial Pβ, we get the coupled equations

α245α2=03+β1+β2α24+3α2+α24+9α2+E2=03β1+β2α24+5α4+E2+1+β1β2α24α2=012β1+β2+β1β2α24+α4+E2=0.E89

Solving these, we find that V0=50, and the three possible eigenvalues, E2,0=2.6301, E2,2=19.0121, and E2,4=43.2490.

5.3 Antisymmetric solutions

In order to find antisymmetric solutions to Eq. (86), we set fβ=sinhxgβ, to obtain

ββ1d2gdx2+αβ2+α+2β12dgdx+α+α24β+α24+α4+E2+14g=0.E90

This CHE can be solved in power series: gβ=g0if N=0, or gβ=g0i=1Nββifor N>0. Then, α=4N+1, and

E=12α2+α4i=1Nβi14N4N24N1.E91

Here, V0=8N+12, and all even and odd solutions have different V0. The maximum solutions order is n=2N+1. For example, for N=3we get α=16, V0=128, and

β1+β2+β33αα24+α24+13α4+E2494=0β1+β2+β3α249α4E2254+β1β2+β2β3+β3β1α242α152=03β1+β2+β3+β1β2+β2β3+β3β1α24+5α4+94+E2+β1β2β3α24+α=012β1β2+β2β3+β3β1β1β2β3α24α4E214=0.E92

We find four eigenvalues, E3,1=12.8152, E3,3=40.4568, E3,5=75.7246, and E3,7=117.003.

6. The potential function Vx=V0sinh4xksinh2x

Now we apply our analysis to the problem with Vx=V0sinh4xksinh2x, which is a symmetric double well if k>0. To find even solutions, we set again βx=cosh2xand ψβ=eα2βfβ, with α2=2V0,

ββ1d2fdβ2+αββ1+β12df
+α2β41+kαβ2+α4+E2α241+kf=0.E93

We now find that V0=22N+121+k, kvarying freely. For example, if N=0, E0,0=1/1+k, and no negative energy eigenvalues may exist. For N=1the two energy eigenvalues found are

E=91+k±1+k2+361+kE94

meaning that for k>3/2we will have negative eigenvalues. Note that for N>0, it is always possible to find a zero-energy ground state, a feature that may have cosmological implications [18].

For the case with N=2, choosing k=4, the energy eigenvalues are E2,0=3.74456, E2,2=1.00000, and E2,4=7.74456. The corresponding eigenfunctions are plotted in Figure 1.

Figure 1.

Left: the three even eigenfunctions (narrow solid lines) found analytically for k=4 and N=2, together with the corresponding eigenvalues (dashed lines). Right: the three odd eigenfunctions (narrow solid lines) found analytically for k=5 and N=2, together with the corresponding eigenvalues (dashed lines). The unsolved eigenvalues are shown in dotted lines.

Now, to find the antisymmetric eigenfunctions, we set fβ=sinhxgβ, to get the CHE

ββ1d2gdβ2+αβ2+α+2β12dg+βα241+cα+α4+E2α241+c+14g=0.E95

For N=0we get that α=4/1+kand E1=6/1+k1/2, such that if k>11, we may find negative energy eigenvalues. For N=2, α=12/1+k, if we set k=5, the energy eigenvalues found are E2,1=7.11693, E2,3=1.08119, and E2,5=9.53574. The eigenfunctions are plotted in Figure 1.

Note that in this case E1E0/E0=0.0052, and it is not possible to distinguish these eigenvalue’s lines from each other in Figure 1 for antisymmetric eigenvalues, implying quasi-degenerate eigenstates. A similar effect is seen in the symmetric case.

6.1 The case with k=1

As was seen in Section VI, the ground-state energy diverges as 1/1+kand as k1, and this also happens to all higher-order even eigenvalues (see Eq. (94)). This is a strange behavior, 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 V0is also divergent when k1.

6.2 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 method [25].

Indeed, let us consider Schrödinger’s problem with the potential function

Vx=α22cosh2x3α2coshx+αcoshx.E96

For this problem, the ground-state eigenfunction and eigenvalue are given by

ψ=ψ0eαcoshxcoshx,E=α212E97

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.

Acknowledgments

This work was partially supported by CONACYT 179881 grants and PROMEP grants UGTO-CA-3. This work is part of the collaboration within the Instituto Avanzado de Cosmología. E. Condori-Pozo is supported by a CONACYT graduate fellowship.

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José Socorro García Díaz, Marco A. Reyes, Carlos Villaseñor Mora and Edgar Condori Pozo (March 15th 2019). Supersymmetric Quantum Mechanics: Two Factorization Schemes and Quasi-Exactly Solvable Potentials [Online First], IntechOpen, DOI: 10.5772/intechopen.82254. Available from:

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