Quasi-exactly solvable extended trigonometric Pöschl-Teller potentials with position-dependent mass
C. Quesne

=22pt plus 1pt minus 1pt

TrIK-SVM : an alternative decomposition for kernel methods in Kreĭn spaces
    Gaëlle Loosli^1,2









1- PobRun 

Brioude, France 




2- UCA - LIMOS UMR 6158 CNRS  

Clermont-Ferrand, France


    December 30, 2023
==========================================================================================================================



 
Infinite families of quasi-exactly solvable position-dependent mass Schrödinger equations with known ground and first excited states are constructed in a deformed supersymmetric background. The starting points consist in one- and two-parameter trigonometric Pöschl-Teller potentials endowed with a deformed shape invariance property and, therefore, exactly solvable. Some extensions of them are considered with the same position-dependent mass and dealt with by a generating function method. The latter enables to construct the first two superpotentials of a deformed supersymmetric hierarchy, as well as the first two partner potentials and the first two eigenstates of the first potential from some generating function W_+(x) [and its accompanying function W_-(x)]. The generalized trigonometric Pöschl-Teller potentials so obtained are thought to have interesting applications in molecular and solid state physics.


Short title: Extended trigonometric Pöschl-Teller potentials

PACS Nos.: 03.65.Fd, 03.65.Ge





§ INTRODUCTION


The trigonometric Pöschl-Teller (TPT) potential (also called Pöschl-Teller I or Darboux-Pöschl-Teller potential) is one of the most valuable exactly solvable (ES) potentials in nonrelativistic quantum mechanics <cit.>. It is indeed close to potentials widely used in molecular physics to describe out-of-plane bending vibrations and in solid state physics to provide models for one-dimensional crystals <cit.>. It is also related to the Scarf I potential <cit.> via simple changes of variable and of parameters <cit.>.


The TPT potential is (translationally) shape invariant (SI) in supersymmetric (SUSY) quantum mechanics <cit.>. Such a property provides an easy way of solving the corresponding Schrödinger equation <cit.>. First- and second-order SUSY transformations have been used to generate new potentials whose spectrum slightly differs from the TPT one <cit.>. Recently, some extensions of the TPT potential have been extensively studied (see, e.g., <cit.> and references quoted therein) in connection with the new concepts of exceptional orthogonal polynomials <cit.>, para-Jacobi polynomials <cit.>, or confluent Darboux transformations <cit.>.


On the other hand, considering a position-dependent mass (PDM) instead of a constant one in the Schrödinger equation is known to play an important role in many physical problems, such as the study of electronic properties of semiconductor heterostructures <cit.>, quantum wells and quantum dots <cit.>, helium clusters <cit.>, graded crystals <cit.>, quantum liquids <cit.>, metal clusters <cit.>, nuclei <cit.>, nanowire structures <cit.>, and neutron stars <cit.>.


Exact solutions of PDM Schrödinger equations may provide a conceptual understanding of some physical phenomena, as well as a testing ground for some approximation schemes. Such solutions may belong not only to ES Schrödinger equations, for which all the eigenstates can be found explicitly by algebraic means, but also to quasi-exactly solvable (QES) equations, for which only a finite number of eigenstates can be derived in this way for some ad hoc couplings, while the remaining ones can only be obtained through numerical calculations.


The generation of PDM and potential pairs leading to such exact solutions has been achieved by various methods (see, e.g., <cit.> and references quoted therein). In particular, on taking advantage of the known equivalence of PDM problems to those arising from a deformation of the canonical commutation relations <cit.>, it has been shown that several well-known ES potentials in a constant mass background remain ES for a well chosen PDM <cit.>. This has been achieved by using a deformed supersymmetric (DSUSY) approach and a deformed shape invariance (DSI) concept. Among those potentials, one finds both the one- and two-parameter TPT potentials.


The aim of the present work is to construct infinite families of QES extensions of these ES PDM and TPT potential pairs with known ground and first excited states. For such a purpose, we plan to use a recently devised generating function method <cit.> (see also <cit.>), generalizing a procedure known for constant mass problems <cit.>.


This paper is organized as follows. In sec. 2, the description of PDM Schrödinger equations in DSUSY and the DSI property are reviewed, then the corresponding results for the ES one- and two-parameter TPT potentials are recalled. In sec. 3, the generating function method for constructing PDM Schrödinger equations with known ground and first excited states is presented. Such a procedure is then applied to extensions of one- and two-parameter TPT potentials in secs. 4 and 5, respectively. Finally, sec. 6 contains the conclusion.




§ DEFORMED SUPERSYMMETRIC APPROACH TO THE TRIGONOMETRIC PÖSCHL-TELLER POTENTIALS WITH POSITION-DEPENDENT MASS


The standard Schrödinger equation


    (p̂^2 + V(x) - E) ψ(x) = 0,


where p̂ = -  i d/dx and ħ = 1, is known to be ES for the one- and two-parameter TPT potentials <cit.>, defined by


    V(x) = A(A-1) ^2 x,      - π2 < x < π2,      A>1,


and


    V(x) = A(A-1) ^2 x + B(B-1) ^2 x,      0 < x < π2,      A, B>1,


respectively.


Let us replace p̂ by π̂ = -  i√(f(x)) (d/dx) √(f(x)), where f(x) is some positive and smooth parameter-dependent function and π̂ is assumed to be Hermitian with respect to the measure dx <cit.>. Then the standard commutation relation [x̂, p̂] =  i is changed into [x̂, π̂] =  i f(x) and the conventional Schrödinger equation (<ref>) becomes


    (Ĥ - E) ψ(x)    = (π̂^2 + V(x) - E) ψ(x) 
       = (- √(f(x))d/dx f(x) d/dx√(f(x)) + V(x) - E) ψ(x) = 0.


This deformed Schrödinger equation can be interpreted as a PDM one,


    (- m^-1/4(x) d/dx m^-1/2(x) d/dx m^-1/4(x) + V(x) - E) ψ(x) = 0,


where m(x) = 1/f^2(x). As is well known, the noncommutativity of m(x) with the differential operator d/dx creates an ordering ambiguity in PDM Schrödinger equations <cit.>. The ordering obtained in (<ref>) is that chosen by Mustafa and Mazharimousavi <cit.>, from which other orderings can be taken care of by replacing V(x) by some effective potential V_ eff(x) including derivatives of m(x).


Bound state wavefunctions ψ_n(x) of eq. (<ref>) (or, equivalently, (<ref>)) have to be square integrable on the interval of definition (x_1, x_2) of V(x) with respect to the measure dx and, in addition, must ensure the Hermiticity of Ĥ or, equivalently, that of π̂, imposing that <cit.>


    |ψ_n(x)|^2 f(x) = |ψ_n(x)|^2/√(m(x))→ 0     for x → x_1 and x → x_2.




A DSUSY approach to eq. (<ref>) consists in considering a pair of partner Hamiltonians, defined on the same interval (x_1, x_2),


    Ĥ_1,2 = π̂^2 + V_1,2(x) + E_0,      V_1,2(x) = W^2(x) ∓ f(x) dW/dx,


where E_0 denotes the ground state energy of eq. (<ref>) and V_1(x) is the rescaled potential V_1(x) = V(x) - E_0 <cit.>. The superpotential W(x) in eq. (<ref>) can be expressed in terms of the ground state wavefunction ψ_0(x) of Ĥ_1 through


    W(x) = - f(x) d/dxlogψ_0(x) - 1/2df/dx


or, conversely,


    ψ_0(x) ∝ f^-1/2exp(- ∫^x W(x')/f(x') dx').




The two first-order differential operators


    Â^± = ∓√(f(x))d/dx√(f(x)) + W(x),


allow to rewrite the two partner Hamiltonians (<ref>) as


    Ĥ_1 = Â^+ Â^- + E_0,     Ĥ_2 = Â^- Â^+ + E_0,


so that the latter intertwine with Â^+ and Â^- as Â^- Ĥ_1 = Ĥ_2 Â^- and Â^+ Ĥ_2 = Ĥ_1 Â^+. The ground state wavefunction ψ_0(x) of Ĥ_1 is annihilated by the operator Â^-, while the ground state wavefunction ψ'_0(x) of Ĥ_2 is transformed by Â^+ into the first excited state wavefunction ψ_1(x) of Ĥ_1.


This procedure can in principle be iterated by considering Ĥ_2 as a new starting Hamiltonian, thereby obtaining another DSUSY pair of partner Hamiltonians


    Ĥ'_1,2 = π̂^2 + V'_1,2(x) + E'_0,      V'_1,2(x) = W^' 2(x) ∓ f(x) 
      dW'/dx,


where


    V'_1(x) + E'_0 = V_2(x) + E_0.


Then the first excited state wavefunction of Ĥ_1 with energy E_1 = E'_0,


    ψ_1(x) ∝Â^+ ψ'_0(x),


can be obtained from the ground state wavefunction of Ĥ'_1 = Ĥ_2, given by


    ψ'_0(x) ∝ f^-1/2exp(- ∫^x W'(x')/f(x') dx').




Equation (<ref>) can be rewritten as


    W^2(x) + f(x) dW/dx = W^' 2(x) - f(x) dW'/dx + E_1 - E_0


in terms of the two superpotentials W(x) and W'(x). Such a condition can be satisfied, in particular, whenever, up to some additive constant R, V_1(x) and V_2(x) are similar in shape and differ only in the parameters that appear in them. In such a case, V(x) is said to be deformed shape invariant (DSI) and eq. (<ref>) is referred to as the DSI condition. The latter can then be generalized to any neighbouring members of a DSUSY hierarchy and the whole bound state spectrum of eq. (<ref>) can be easily derived.


For the one-parameter TPT potential (<ref>), the DSI condition is satisfied for the deforming function


    f(x) = 1 + αsin^2 x,      -1 < α 0,


corresponding to a PDM m(x) = (1 + αsin^2 x)^-2, and for the two superpotentials <cit.>


    W(x)    = λtan x,     λ = 12(1 + α + Δ),     Δ = √((1+
          α)^2 + 4A(A-1)), 
     
      W'(x)    = λ' tan x,     λ' = λ + 1 + α.


The first two partner potentials read


    V_1(x)    = A(A-1) ^2 x - A(A-1) - 12(1 + α + Δ), 
    
      V_2(x)    = [A(A-1) + (1+α) (1+α+Δ)] ^2 x - A(A-1) 
         - 12(1+2α) (1+α+Δ),


while the ground and first excited state energies of V(x) are given by


    E_0    = λ(λ-α) = A(A-1) + 12(1+α+Δ), 
    
      E_1    = (λ+1)^2 - α(λ-1) = A(A-1) + 12(5+5α+3Δ).


More generally, the whole bound state spectrum is obtained as


    E_n    = (λ+n)^2 - α(λ-n^2) 
       = A(A-1) + 12(1+α+Δ) + (1+α+Δ)n + (1+α)n^2, 
          n=0, 1, 2, …,


with the corresponding wavefunctions


    ψ_n(x) ∝ f^- 1/2(λ/1+α + 1) (cos x)^λ/1+
         α C_n^(λ/1+α)(t),      t = √(1+α/f)sin x,


expressed in terms of Gegenbauer polynomials <cit.>. Note that, in this case, eq. (<ref>) does not provide any additional condition since it is automatically fulfilled for square integrable functions ψ_n(x) on (-π/2, π/2).


For the two-parameter TPT potential (<ref>), the deforming function, obtained in <cit.>,[The results given in <cit.> are for the Scarf I potential. They have been transformed here for the TPT potential by using the changes of variable and parameters given in <cit.>.] writes


    f(x) = 1 + αcos 2x,      0 < |α| < 1,


with corresponding PDM m(x) = (1+αcos 2x)^-2, and the two superpotentials are


    W(x)    = λtan x - μ x,     λ = 12(1-α+Δ_1),     μ = 12(1+α+Δ_2), 
    Δ_1    = √((1-α)^2 + 4A(A-1)),     Δ_2 = √((1+α)^2 + 4B(B-1)), 
    
      W'(x)    = λ' tan x - μ'  x,     λ' = λ+1-α,     μ' = μ+1+α.


The first two partner potentials read


    V_1(x)    = A(A-1) ^2 x + B(B-1) ^2 x - A(A-1) - B(B-1) - 32(1-α^2) 
         - (1+α)Δ_1 - (1-α)Δ_2 - 12Δ_1Δ_2, 
    
      V_2(x)    = [A(A-1) + (1-α)(1-α+Δ_1)] ^2 x 
          + [B(B-1) + (1+α)(1+α+Δ_2] ^2 x - A(A-1) - B(B-1) 
          - 12(3+5α^2) - (1-α)Δ_1 - (1+α)Δ_2 - 12Δ_1 Δ_2,


while the ground and first excited state energies of V(x) are given by


    E_0    = (λ+μ)^2 + 2α(λ-μ) = A(A-1) + B(B-1) + 32(1-α^2) 
         
          + (1+α)Δ_1 + (1-α)Δ_2 + 12Δ_1Δ_2, 
    
      E_1    = (λ+μ+2)^2 + 6α(λ-μ) - 4α^2 = A(A-1) + B(B-1) 
          + 192(1-α^2) + 3(1+α)Δ_1 + 3(1-α)Δ_2 + 12Δ_1
          Δ_2.
 

The whole bound state spectrum is obtained as 


    E_n    = (λ+μ+2n)^2 + 2α(λ-μ)(2n+1) - 4α^2 n^2 
       = A(A-1) + B(B-1) + 32(1-α^2) + (1+α)Δ_1 + (1-α)Δ_2 + 12Δ_1 Δ_2 
          + 2n[2(1-α^2) + (1+α)Δ_1 + (1-α)Δ_2] + 4(1-α^2)n^2, 
            n=0, 1, 2, …,


with the corresponding wavefunctions <cit.>


    ψ_n(x)    ∝ f^-1/2(1 + λ/1-α + μ/1+α)
          (cos x)^λ/1-α (sin x)^μ/1+α P_n^(μ/1+α -
          1/2, λ/1-α - 1/2)(t), 
    
      t    = cos 2x + α/1 + αcos 2x,


expressed in terms of Jacobi polynomials.




§ GENERATING FUNCTION METHOD FOR PDM SCHRÖDINGER EQUATIONS WITH TWO KNOWN EIGENSTATES



Let us start from eq. (<ref>) relating the superpotentials W(x) and W'(x) of the first two steps of a DSUSY hierarchy and  let us define the two functions <cit.>


    W_+(x) = W'(x) + W(x),      W_-(x) = W'(x) - W(x).


In terms of the latter, eq. (<ref>) can be rewritten as


    f(x) dW_+/dx = W_+(x) W_-(x) + E_1 - E_0.


Hence, W_-(x) can be expressed in terms of W_+(x) and the energy difference E_1-E_0 as


    W_-(x) = f(x) dW_+(x)/dx + E_0 - E_1/W_+(x).




The generating function method starts from two functions W_+(x) and W_-(x) that are compatible, i.e., such that there exists some positive constant E_1-E_0 satisfying eq. (<ref>). The two superpotentials W(x) and W'(x) are then obtained from eq. (<ref>) as W(x) = 1/2(W_+ - W_-) and W'(x) = 1/2(W_+ + W_-). The starting potential V(x) and its ground state energy E_0 are determined from W(x) through eq. (<ref>) and the ground state wavefunction is derived from eq. (<ref>). The knowledge of E_1-E_0 and E_0 provides the first excited state energy E_1, while a combination of eqs. (<ref>), (<ref>), (<ref>), and (<ref>) leads to the corresponding wavefunction


    ψ_1(x)    ∝(- f d/dx - 1/2df/dx + W) f^-1/2exp(-
         ∫^x W'(x')/f(x') dx') 
       ∝ [W'(x) + W(x)] f^-1/2exp(- ∫^x W'(x')/f(x') dx') 
       ∝ W_+(x) f^-1/2exp(- ∫^x W'(x')/f(x') dx').


The construction of the first two bound state wavefunctions ψ_0(x) and ψ_1(x) of V(x) is of course only valid provided such functions satisfy both the square integrability condition on (x_1,x_2) and the additional restriction (<ref>). As observed in sec. 2, the latter is automatically fulfilled for the deforming functions f(x) considered for the one- and two-parameter TPT potentials provided the wavefunctions are square integrable.




§ EXTENSIONS OF THE ONE-PARAMETER TRIGONOMETRIC PÖSCHL-TELLER POTENTIALS



In the present section, we will deal with an infinite family of extensions of the one-parameter TPT potential (<ref>), defined by


    V^(m)(x) = ∑_k=1^2m+1 A_2k^2kx,      - π/2 < x < π/2,     
      m = 1, 2, …,


with A_4m+2>0. As it is obvious, the m=0 case would give back potential (<ref>) with A_2 = A(A-1). Our aim consists in showing that parameters A_2, A_4, …, A_4m can be found in terms of A_4m+2 and α in such a way that the PDM Schrödinger equation (<ref>) with f(x) and V(x) given by (<ref>) and (<ref>), respectively, has known ground and first excited states. The corresponding PDM will then be


    m(x) = (1 + αsin^2x)^-2,      -1 < α 0.












For such a purpose, let us consider the generating functions


    W_+(x)    = 2√(A_4m+2)∑_k=0^m (2m+1)!!/(2k+1)!! (2m-2k)!! (1+α)^k-m
          (tan x)^2k+1, 
    
      W_-(x)    = (2m+1) (1+α) tan x.


To prove their compatibility, we have to show that there exists some positive constant E_1-E_0 satisfying eq. (<ref>). Straightforward calculations show that


    dW_+/dx   = 2√(A_4m+2){(2m+1)!!/(2m)!! (1+α)^-m
          + ∑_k=1^m (2m+1)!!/(2k-1)!! (2m-2k)!! (1+α)^k-m (tan x)^2k}^2 x,
 


    f dW_+/dx   = 2√(A_4m+2){(2m+1)!!/(2m)!! (1+α)^-m
          + (2m+1)∑_k=0^m (2m+1)!!/(2k+1)!! (2m-2k)!! (1+α)^k+1-m 
           (tan x)^2k+2},


and


    W_+ W_-  
       = 2√(A_4m+2) (2m+1)∑_k=0^m (2m+1)!!/(2k+1)!! (2m-2k)!! 
      (1+α)^k+1-m (tan x)^2k+2,


so that we indeed get


    E_1-E_0 = 2√(A_4m+2)(2m+1)!!/(2m)!! (1+α)^-m > 0.




From (<ref>) and (<ref>), we then obtain the two superpotentials W(x) and W'(x) in the form


    W(x) = ∑_k=0^m λ_k (tan x)^2k+1,      W'(x) = ∑_k=0^m λ'_k 
      (tan x)^2k+1,


where

    λ_0    = √(A_4m+2)(2m+1)!!/(2m)!! (1+α)^-m - 1/2(2m+1)(1+α), 
    λ'_0    = λ_0 + (2m+1)(1+α), 
    λ_k    = λ'_k = √(A_4m+2)(2m+1)!!/(2k+1)!! (2m-2k)!! (1+α)^k-m,     
         k=1, 2, …, m.
    .




To determine V(x) and E_0 from eqs. (<ref>), (<ref>), and (<ref>), it is convenient to proceed in two steps: first to express V_1(x) as an expansion in tan^2 x, 


    V_1(x) = ∑_k=0^2m+1 a_k (tan x)^2k,


then to reexpress it as an expansion in ^2 x by making use of the relation tan^2 x = ^2 x -1. In such a way, we obtain E_0 and the parameters A_2k, k=1, 2, …, 2m of eq. (<ref>) as


    E_0 = ∑_k=0^2m+1 (-1)^k+1 a_k,      A_2k = ∑_l=k^2m+1 (-1)^l-klk
      a_l.


From the values of the coefficients a_k in eq. (<ref>), we get


    E_0    = 1/4 (2m+1) (1+α) [2m+1 + (2m+3)α] 
          + √(A_4m+2)(2m+1)!!/(2m)!! (1+α)^-m
         ×[1 + 2(2m+1) ∑_k=1^m+1 (-1)^k 
          (2m)!!/(2k-1)!! (2m-2k+2)!! (1+α)^k] 
          - A_4m+2 (1+α)^-2m[∑_k=1^m+1 (-1)^k S^(m,k)_0,k-1 (1+
          α)^k-1
          + ∑_k=m+2^2m+1 (-1)^k S^(m,k)_k-m-1,m (1+α)^k-1],



    A_2    = 1/4(2m+1)(2m+3) (1+α)^2 
          + 2(2m+1) √(A_4m+2)∑_k=1^m+1 (-1)^k k (2m+1)!!/(2k-1)!! (2m-2k+2)!! 
          (1+α)^k-m
          - A_4m+2 (1+α)^-2m[∑_k=1^m+1 (-1)^k k S^(m,k)_0,k-1
          (1+α)^k-1
          +∑_k=m+2^2m+1 (-1)^k k S^(m,k)_k-m-1,m (1+α)^k-1],



    A_2k   = - 2(2m+1) √(A_4m+2)∑_l=k^m+1 (-1)^l-klk(2m+1)!!/(2l-1)!! (2m-2l+2)!! (1+α)^l-m
          + A_4m+2 (1+α)^-2m[∑_l=k^m+1 (-1)^l-klk S^(m,l)_0,l-1
          (1+α)^l-1
          + ∑_l=m+2^2m+1 (-1)^l-klk S^(m,l)_l-m-1,m (1+α)^l-1], 
               2 ≤ k ≤ m+1,



    A_2k = A_4m+2∑_l=k^2m+1 (-1)^l-klk S^(m,l)_l-m-1,m (1+α)^l-2m-1,
           m+2 ≤ k ≤ 2m,


where we have introduced finite sums S^(m,k)_a,b, 0 ≤ a ≤ b ≤ m, defined by


    S^(m,k)_a,b = ∑_l=a^b [(2m+1)!!]^2/(2l+1)!! (2k-2l-1)!! (2m-2l)!! (2m-2k+2l+2)!!.


Equations (<ref>) and (<ref>) yield the first excited state energy E_1.


It remains to determine the wavefunctions ψ_0(x) and ψ_1(x) from eqs. (<ref>) and (<ref>), respectively. For such a purpose, it is useful to express the ratios W(x)/f(x) and W'(x)/f(x) in terms of a new variable y=cos x. For the former, for instance, we get the relation


    W(x)/f(x)   = sin x [y^2m+1 (1+α-α y^2)]^-1∑_k=0^m λ_k
           y^2m-2k (1-y^2)^k 
       = sin x [y^2m+1 (1+α-α y^2)]^-1∑_p=0^m [∑_l=0^p (-1)^l
           m+l-plλ_m+l-p] y^2p
       = sin x (∑_κ=0^m C_2κ+1/y^2κ+1 
           + α C_1 y/1+α-α y^2),


with 

    C_2κ+1   = 1/(1+α)^m+1-κ∑_p=0^m-κα^m-κ-p
           (1+α)^p ∑_l=0^p (-1)^l l+m-plλ_l+m-p, 
           κ=0, 1, …, m,


from which the integration in eq. (<ref>) is straightforward. The results read


    ψ_0(x) ∝ f^-1/2(C_1+1) (cos x)^C_1exp(-∑_κ=1^m 
      C_2κ+1/2κ^2κx)


and


    ψ_1(x)    ∝ f^-1/2(C_1+2m+2) (cos x)^C_1exp(-∑_κ=1^m 
         C_2κ+1/2κ^2κx) 
         ×∑_k=0^m {[∑_l=0^k (-1)^k-lm-lk-l(2m+1)!!/(2l+1)!! (2m-2l)!! (1+α)^l-m] 
         ×sin^2k+1x}.


The function f(x) having a finite value 1+α for x →±π/2, the behaviour of ψ_0(x) and ψ_1(x) at the interval boundaries is determined by that of exp[- C_2m+1^2mx/(2m)],
where C_2m+1= √(A_4m+2)/(1+α) > 0, thereby showing that such functions are square integrable, as it should be.


As an illustration, let us present some detailed results for the m=1 case. The potential reads


    V^(1)(x)    = [15/4 (1+α)^2 + 3(1+4α) √(A_6) 
         + 3(4α^2-1)/4(1+α)^2 A_6] ^2 x 
          - 3√(A_6)[2(1+α) + α/1+α√(A_6)] ^4 x
         + A_6 ^6 x.


Its ground and first excited state energies are


    E_0    = 3/4(1+α)(3+5α) + 3(-1+2α+4α^2)/2(1+α)√(A_6)
          + (1-2α)^2/4(1+α)^2 A_6, 
    
      E_1    = 3/4(1+α)(3+5α) + 3(1+2α+4α^2)/2(1+α)√(A_6)
          + (1-2α)^2/4(1+α)^2 A_6,


with corresponding wavefunctions


    ψ_0(x)    ∝ f^1/4 - √(A_6)/4(1+α)^2 
          (cos x)^√(A_6)/2(1+α)^2 - 3/2exp(- √(A_6)/2(1+α)^2 x),  
    ψ_1(x)    ∝ f^-5/4 - √(A_6)/4(1+α)^2 
          (cos x)^√(A_6)/2(1+α)^2 - 3/2sin x [3 - (1-2α)sin^2 x] 
         ×exp(- √(A_6)/2(1+α)^2 x).


As can be checked, the odd wavefunction ψ_1(x) has a single zero at x=0 in the defining interval (-π/2, π/2), since no real x satisfies the condition sin^2 x = 3/(1-2α) > 1.










In fig. 1, an example of extended potential (<ref>) is plotted. Its corresponding (rescaled) unnormalized wavefunctions (<ref>) and (<ref>) are displayed in fig. 2.




§ EXTENSIONS OF THE TWO-PARAMETER TRIGONOMETRIC PÖSCHL-TELLER POTENTIAL



Let us consider next an infinite family of extensions of the two-parameter TPT potential (<ref>), defined by


    V^(m_1,m_2)(x) = ∑_k=1^2m_1+1 A_2k^2kx + ∑_l=1^2m_2+1 B_2l^2lx,
           0 < x < π/2,


where m_1 and m_2 are two nonnegative integers and A_4m_1+2, B_4m_2+2 > 0. For m_1 = m_2 = 0, eq. (<ref>) would give back eq. (<ref>) with A_2 = A(A-1) and B_2 = B(B-1). Here, we wish to determine parameters A_2, A_4, …, A_4m_1, B_2, B_4, … B_4m_2 in terms of A_4m_1+2, B_4m_2+2, and α in such a way that eq. (<ref>) with f(x) and V(x) given by (<ref>) and (<ref>), respectively, has known ground and first excited states. The corresponding PDM will then be


    m(x) = (1 + αcos2x)^-2,      0 < |α| <1.


It is enough to assume m_1 ≥ m_2, because the m_1<m_2 case could be easily obtained from that with m_1>m_2 by permuting the roles of ^2x and ^2x, which can be achieved by the change of variable x →π/2 - x and the change of parameter α→ - α. Since the cases m_1 ≥ m_2 > 0 and m_1 > m_2=0 have to be distinguished, we will start with the former, then point out the changes to be made to care for the latter.




 §.§ Extensions with m_1 ≥ m_2 > 0


Let us consider the generating functions


    W_+(x)    = 2√(A_4m_1+2)∑_k=0^m_1m_1+m_2+1m_2+k+1(1+α/1-α)^m_1-k (tan x)^2k+1
          - 2√(B_4m_2+2)∑_l=0^m_2m_1+m_2+1m_1+l+1(1-α/1+α)^m_2-l ( x)^2l+1, 
    
      W_-(x)    = (2m_1+1) (1-α) tan x - (2m_2+1) (1+α)  x.


A calculation similar to that carried out in sec. 4 shows that such functions are compatible and that E_1-E_0 is given by


    E_1-E_0    = 4 (m_1+m_2+1)!/m_1! m_2![√(A_4m_1+2)(1+α)^m_1+1/(1-α)^m_1
          + √(B_4m_2+2)(1-α)^m_2+1/(1+α)^m_2].












The two superpotentials W(x) and W'(x) can now be expressed as


    W(x)    = ∑_k=0^m_1λ_k (tan x)^2k+1 - ∑_l=0^m_2μ_l ( x)^2l+1, 
    
      W'(x)    = ∑_k=0^m_1λ'_k (tan x)^2k+1 - ∑_l=0^m_2μ'_l ( x)^2l+1,


where


    λ_0   = √(A_4m_1+2)m_1+m_2+1m_2+1(1+α/1-α)^m_1 - (m_1+1/2)(1-α), 
    λ'_0    = λ_0 + (2m_1+1)(1-α), 
    λ_k    = λ'_k = √(A_4m_1+2)m_1+m_2+1m_2+k+1(1+α/1-α)^m_1-k,      k=1, 2, …, m_1, 
    μ_0   = √(B_4m_2+2)m_1+m_2+1m_1+1(1-α/1+α)^m_2 - (m_2+1/2)(1+α), 
    μ'_0    = μ_0 + (2m_2+1)(1+α), 
    μ_l    = μ'_l = √(B_4m_2+2)m_1+m_2+1m_1+l+1(1-α/1+α)^m_2-l,      l=1, 2, …, m_2.




As in sec. 4, the determination of E_0 and A_2, A_4, …, A_4m_1, B_2, B_4, …, B_4m_2 in terms of A_4m_1+2, B_4m_2+2, and α can be carried out in two steps: first to obtain the coefficients a_k and b_l in the expansion


    V_1(x) = ∑_k=0^2m_1+1 a_k (tan x)^2k + ∑_l=1^2m_2+1 b_l ( x)^2l,


then to express the searched for quantities as


    E_0    = ∑_k=0^2m_1+1 (-1)^k+1 a_k + ∑_l=1^2m_2+1 (-1)^l+1 b_l, 
    
      A_2k   = ∑_l=k^2m_1+1 (-1)^l-klk a_l,      k=1, 2, …, 2m_1, 
    
      B_2l   = ∑_k=l^2m_2+1 (-1)^k-lkl b_k,      l=1, 2, …, 2m_2.


After some lengthy, but straightforward calculations, we get the results detailed in appendix A. 


To determine the wavefunctions, we rewrite this time W/f and W'/f in terms of the variable y=cos 2x. For the former, for instance, we get


    W(x)/f(x) = sin 2x (∑_p=1^m_1+1C_p/(1+y)^p - ∑_q=1^m_2+1D_q/(1-y)^q - α (C_1+D_1)/1+α y),


with


    C_p = ∑_q=p-1^m_1 2^q (-α)^q-p+1/(1-α)^q-p+2∑_k=q^m_1 (-1)^k-qkqλ_k,      p=1, 2, …,m_1+1,


and

    D_q = ∑_p=q-1^m_2 2^p α^p-q+1/(1+α)^p-q+2∑_l=p^m_2 (-1)^l-plpμ_l,      q=1, 2, …, m_2+1.


The results read


    ψ_0(x)    ∝ f^- 1/2(C_1+D_1+1) (cos x)^C_1 (sin x)^D_1
         ×exp[- ∑_p=2^m_1+1C_p/2^p (p-1) ( x)^2(p-1)
          - ∑_q=2^m_2+1D_q/2^q (q-1) ( x)^2(q-1)]


and


    ψ_1(x)    ∝ f^- 1/2(C_1+D_1+2m_1+2m_2+3) (cos x)^C_1 (sin x)^D_1
         ×{-2 √(B_4m_2+2)∑_k=0^m_2 (-1)^k m_1+m_2+1k(2α/1+α)^k sin^2kx 
          + ∑_k=m_2+1^m_1+m_2+1[m_1+m_2+1k
         ×(2 √(A_4m_1+2)(1+α/1-α)^m_1+m_2-k+1
          F(k-m_2-1,k;:1+α/1-α) 
          - 2√(B_4m_2+2) (-1)^k F(m_2,k; 1-α/1+α))
          sin^2kx] }
         ×exp[- ∑_p=2^m_1+1C_p/2^p (p-1) ( x)^2(p-1)
          - ∑_q=2^m_2+1D_q/2^q (q-1) ( x)^2(q-1)],

 
 where we have defined
 
 
    F(n,k;z) = ∑_p=0^n (-1)^p kp z^p,      k>n.


At the boundaries x=0 and x=π/2 of the defining interval, the behaviour of ψ_0(x) and ψ_1(x) is governed by that of exp[-D_m_2+1 ( x)^2m_2/(2^m_2+1m_2)] and exp[-C_m_1+1 ( x)^2m_1/(2^m_1+1m_1)], where D_m_2+1 = 2^m_2√(B_4m_2+2)/(1+α) > 0 and C_m_1+1 = 2^m_1√(A_4m_1+2)/(1-α) > 0, respectively. Hence, such functions are square integrable on (0,π/2), as it should be.


For the simplest case corresponding to m_1=m_2=1, we get, for instance, the potential


    V^(1,1)(x)    = [15/4(1-α)^2 - 24α√(A_6) + 
          12α(1+2α)/(1-α)^2 A_6 - 6(1-α)/1+α√(A_6B_6)]
          ^2 x 
          + 3√(A_6)[-2(1-α) + 1+3α/1-α√(A_6)] ^4 x + A_6 
          ^6 x 
          + [15/4(1+α)^2 + 24α√(B_6)  
          - 6(1+α)/1-α√(A_6B_6) - 12α(1-2α)/(1+α)^2 B_6 ]
          ^2 x 
          + 3√(B_6)[-2(1+α) + 1-3α/1+α√(B_6)] ^4 x + B_6 
          ^6 x.


Its ground and first excited state energies are given by


    E_0    = 3(2α^2+3) + 12(α^2-2α-1)/1-α√(A_6) + 
          12(α^2+2α-1)/1+α√(B_6)
          + 4(1+2α)^2/(1-α)^2 A_6 + 8(1-2α)(1+2α)/(1-α)(1+α)√(A_6B_6) + 4(1-2α)^2/(1+α)^2 B_6


and


    E_1    = 3(2α^2+3) + 12(3α^2+2α+1)/1-α√(A_6) + 
          12(3α^2-2α+1)/1+α√(B_6)
          + 4(1+2α)^2/(1-α)^2 A_6 + 8(1-2α)(1+2α)/(1-α)(1+α)√(A_6B_6) + 4(1-2α)^2/(1+α)^2 B_6,


with corresponding wavefunctions


    ψ_0(x)    ∝ f^-1+α/(1-α)^2√(A_6) -1-α/(1+α)^2√(B_6)
          +1 (cos x)^2(1+α)/(1-α)^2√(A_6) - 3/2 
          (sin x)^2(1-α)/(1+α)^2√(B_6) - 3/2
         ×exp[- √(A_6)/2(1-α)^2x - √(B_6)/2(1+α)^2x]


and


    ψ_1(x)    ∝ f^-1+α/(1-α)^2√(A_6) -1-α/(1+α)^2√(B_6)
          -2 (cos x)^2(1+α)/(1-α)^2√(A_6) - 3/2 
          (sin x)^2(1-α)/(1+α)^2√(B_6) - 3/2
         ×{- 2√(B_6) + 12α/1+α√(B_6)sin^2x + 6 [
          1+α/1-α√(A_6) + 1-3α/1+α√(B_6)] sin^4x 
          - 4[1+2α/1-α√(A_6) + 1-2α/1+α√(B_6)]
          sin^6x}
         ×exp[- √(A_6)/2(1-α)^2x - √(B_6)/2(1+α)^2x].




In fig. 3, an example of extended potential (<ref>) is plotted. Its corresponding (rescaled) unnormalized wavefunctions (<ref>) and (<ref>) are displayed in fig. 4.




 §.§ Extensions with m_1> m_2=0


For m_1> m_2=0, the results presented in sec. 5.1 remain valid provided we replace √(B_2) by 1 + α + 1/2Δ, Δ = √((1+α)^2 + 4B_2), in the generating functions (<ref>) and (<ref>), which therefore become


    W_+(x)    = 2√(A_4m_1+2)∑_k=0^m_1m_1+1k+1(1+α/1-α)^m_1-k (tan x)^2k+1
          - (2+2α+Δ)  x, 
    
      W_-(x)    = (2m_1+1)(1-α) tan x - (1+α)  x,


with corresponding E_1-E_0 given by


    E_1 - E_0 = 4(m_1+1) (1+α)^m_1+1/(1-α)^m_1√(A_4m_1+2) 
      + 2(m_1+1)(1-α)(2+2α+Δ).




We shall not present the general results, but instead show the simplest example corresponding to m_1=1. In such a case, the potential reads


    V^(1,0)(x)    = [15/4(1-α)^2 - (24α+Δ) √(A_6) + 
          -1+2α+15α^2/(1-α)^2 A_6] ^2x 
          + √(A_6)[-6(1-α) + 1+7α/1-α√(A_6)] ^4x
          + A_6 ^6x + B_2 ^2x,


with ground and first excited state energies


    E_0    = 5/4(1-α)(1-5α) - (1-α)Δ + 
         -2-4α+22α^2+(1+3α)Δ/1-α√(A_6)
          + (1+3α/1-α)^2 A_6 + B_2, 
    
      E_1    = 1/4(1-α)(37+7α) + 3(1-α)Δ + 
         6(1+2α+5α^2)+(1+3α)Δ/1-α√(A_6)
          + (1+3α/1-α)^2 A_6 + B_2,


and corresponding wavefunctions


    ψ_0(x)    ∝ f^-1+α/2(1-α)^2√(A_6) - Δ/4(1+α)
          (cos x)^1+α/(1-α)^2√(A_6) - 3/2 (sin x)^Δ/2(1+α)
          +1/2
         ×exp[- √(A_6)/2(1-α)^2x],  
    ψ_1(x)    ∝ f^-1+α/2(1-α)^2√(A_6) - Δ/4(1+α) - 2
          (cos x)^1+α/(1-α)^2√(A_6) - 3/2 (sin x)^Δ/2(1+α)
          +1/2
         ×{2+2α+Δ - 2[2(1+α)/1-α√(A_6) +2+2α
          +Δ] sin^2x 
          + [2(1+3α)/1-α√(A_6) +2+2α+Δ] sin^4x}exp[- √(A_6)/2(1-α)^2x].




In fig. 5, an example of extended potential (<ref>) is plotted. Its corresponding (rescaled) unnormalized wavefunctions (<ref>) and (<ref>) are displayed in fig. 6.




§ CONCLUSION


In the present paper, we have shown that it is possible to generate infinite families of PDM Schrödinger equations with known ground and first excited states in DSUSY by considering extensions of both one- and two-parameter TPT potentials endowed with a DSI property. If needed, higher energy levels should be calculated numerically. This work completes a previous study <cit.>, where only extensions of some simpler potentials were explicitly constructed, and demonstrates the efficiency of the method proposed there to deal with more complex potentials. This opens the way for building extensions of other potentials with a DSI property, whose treatment is rather involved, such as the Eckart and Rosen-Morse I potentials considered in <cit.>.


Taking into account the usefulness of the TPT potential as a first approximation in several problems of molecular and solid state physics, it is obvious that the exact results presented here for potentials including some extra terms may find helpful applications in such fields. The search for such applications would be another interesting topic for future investigation.




§ APPENDIX A. GENERAL RESULTS FOR EXTENSIONS OF THE TWO-PARAMETER TRIGONOMETRIC PÖSCHL-TELLER POTENTIAL WITH M_1 ≥ M_2 > 0





In this appendix, we present the general results obtained for the ground state energy and the parameters of the extensions of the two-parameter trigonometric Pöschl-Teller potential with m_1 ≥ m_2 > 0:


    E_0    = (m_1+m_2+1)^2 - 2α(m_1-m_2)(m_1+m_2+2) + α^2[(m_1-m_2)^2 
          + 2(m_1+m_2+1)] 
          - √(A_4m_1+2){2m_2 m_1+m_2+1m_2+1(1+α)^m_1+1/(1-α)^m_1
          + ∑_k=1^m_1+1 (-1)^k [(2m_2-2k) m_1+m_2+1m_2+k+1 
           - (2m_1+2k) m_1+m_2+1m_2+k] 
         ×(1+α)^m_1-k+1/(1-α)^m_1-k}
          - √(B_4m_2+2){2m_1 m_1+m_2+1m_1+1(1-α)^m_2+1/(1+α)^m_2
          + ∑_l=1^m_2+1 (-1)^l [(2m_1-2l) m_1+m_2+1m_1+l+1 
           - (2m_2+2l) m_1+m_2+1m_1+l] 
         ×(1-α)^m_2-l+1/(1+α)^m_2-l}
          - A_4m_1+2∑_k=1^2m_1+1[(-1)^k (1+α/1-α)^2m_1-k+1
         ×∑_l= max(0,k-m_1-1)^ min(k-1,m_1)m_1+m_2+1m_2+l+1m_1+m_2+1m_2+k-l] 
          + 2√(A_4m_1+2B_4m_2+2){∑_k=0^m_1[(-1)^k 
          (1+α/1-α)^m_1-m_2-k
         ×∑_l=k^ min(m_2+k,m_1)m_1+m_2+1m_2+l+1m_1+m_2+1m_2+k-l] 
          + ∑_l=1^m_2[(-1)^l 
          (1+α/1-α)^m_1-m_2+l∑_k=l^m_2m_1+m_2+1m_1+k+1m_1+m_2+1m_1+l-k]}
          - B_4m_2+2∑_l=1^2m_2+1[(-1)^l (1-α/1+α)^2m_2-l+1
         ×∑_k= max(0,l-m_2-1)^ min(l-1,m_2)m_1+m_2+1m_1+k+1m_1+m_2+1m_1+l-k],



    A_2    = (m_1+1/2) (m_1+3/2) (1-α)^2 
          - √(A_4m_1+2)∑_k=1^m_1+1{(-1)^k k (1+α)^m_1-k+1/(1-α)^m_1-k
         ×[(2m_2-2k) m_1+m_2+1m_2+k+1 - (2m_1+2k) 
          m_1+m_2+1m_2+k]}
          - A_4m_1+2∑_k=1^2m_1+1[(-1)^k k (1+α/1-α)^2m_1-k+1
         ×∑_l= max(0,k-m_1-1)^ min(k-1,m_1)m_1+m_2+1m_2+l+1m_1+m_2+1m_2+k-l] 
          + 2√(A_4m_1+2B_4m_2+2)∑_k=1^m_1[(-1)^k k 
          (1+α/1-α)^m_1-m_2-k
         ×∑_l=k^ min(m_2+k,m_1)m_1+m_2+1m_2+l+1m_1+m_2+1m_2+k-l],



    A_2k   = √(A_4m_1+2)∑_l=k^m_1+1{(-1)^l-klk(1+α)^m_1-l+1/(1-α)^m_1-l
         ×[(2m_2-2l) m_1+m_2+1m_2+l+1 - (2m_1+2l) 
          m_1+m_2+1m_2+l]}
          + A_4m_1+2∑_l=k^2m_1+1[(-1)^l-klk(1+α/1-α)^2m_1-l+1
         ×∑_p= max(0,l-m_1-1)^ min(l-1,m_1)m_1+m_2+1m_2+p+1m_1+m_2+1m_2+l-p] 
          - 2√(A_4m_1+2B_4m_2+2)∑_l=k^m_1[(-1)^l-klk(1+α/1-α)^m_1-m_2-l
         ×∑_p=l^ min(m_2+l,m_1)m_1+m_2+1m_2+p+1m_1+m_2+1m_2+l-p],       2 ≤ k ≤ m_1+1,



    A_2k   = A_4m_1+2∑_l=k^2m_1+1[(-1)^l-klk(1+α/1-α)^2m_1-l+1
         ×∑_p=l-m_1-1^m_1m_1+m_2+1m_2+p+1m_1+m_2+1m_2+l-p],       m_1+2 ≤ k ≤ 2m_1,



    B_2    = (m_2+1/2) (m_2+3/2) (1+α)^2 
          - √(B_4m_2+2)∑_k=1^m_2+1{(-1)^k k (1-α)^m_2-k+1/(1+α)^m_2-k
         ×[(2m_1-2k) m_1+m_2+1m_1+k+1 - (2m_2+2k) 
          m_1+m_2+1m_1+k]}
          + 2√(A_4m_1+2B_4m_2+2)∑_k=1^m_2[(-1)^k k 
          (1+α/1-α)^m_1-m_2+k
         ×∑_l=k^m_2m_1+m_2+1m_1+l+1m_1+m_2+1m_1+k-l] 
          - B_4m_2+2∑_k=1^2m_2+1[(-1)^k k (1-α/1+α)^2m_2-k+1
         ×∑_l= max(0,k-m_2-1)^ min(k-1,m_2)m_1+m_2+1m_1+l+1m_1+m_2+1m_1+k-l],



    B_2l   = √(B_4m_2+2)∑_k=l^m_2+1{(-1)^k-lkl(1-α)^m_2-k+1/(1+α)^m_2-k
         ×[(2m_1-2k) m_1+m_2+1m_1+k+1 - (2m_2+2k) 
          m_1+m_2+1m_1+k]}
          - 2√(A_4m_1+2B_4m_2+2)∑_k=l^m_2[(-1)^k-lkl(1+α/1-α)^m_1-m_2+k
         ×∑_p=k^m_2m_1+m_2+1m_1+p+1m_1+m_2+1m_1+k-p] 
          + B_4m_2+2∑_k=l^2m_2+1[(-1)^k-lkl(1-α/1+α)^2m_2-k+1
         ×∑_p= max(0,k-m_2-1)^ min(k-1,m_2)m_1+m_2+1m_1+p+1m_1+m_2+1m_1+k-p],     2 ≤ l ≤ m_2+1,




    B_2l   = B_4m_2+2∑_k=l^2m_2+1[(-1)^k-lkl(1-α/1+α)^2m_2-k+1
         ×∑_p=k-m_2-1^m_2m_1+m_2+1m_1+p+1m_1+m_2+1m_1+k-p],       m_2+2 ≤ l ≤ 2m_2.


Note that E_1 can be easily obtained from (<ref>) and (<ref>).




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