Deformed shape invariance and exactly solvable Hamiltonians with position-dependent effective mass

Abstract

Known shape-invariant potentials for the constant-mass Schrödinger equation are taken as effective potentials in a position-dependent effective mass (PDEM) one. The corresponding shape-invariance condition turns out to be deformed. Its solvability imposes the form of both the deformed superpotential and the PDEM. A lot of new exactly solvable potentials associated with a PDEM background are generated in this way. A novel and important condition restricting the existence of bound states whenever the PDEM vanishes at an end point of the interval is identified. In some cases, the bound-state spectrum results from a smooth deformation of that of the conventional shape-invariant potential used in the construction. In others, one observes a generation or suppression of bound states, depending on the mass-parameter values. The corresponding wavefunctions are given in terms of some deformed classical orthogonal polynomials.

Point Canonical Transformation versus Deformed Shape Invariance for Position-Dependent Mass Schrödinger Equations

“…with a mass and an effective potential given by 5) respectively [7,15,16]. Here a prime stands for derivative with respect to x.…”

“…The DSI method [15] considers the Hamiltonian H (α) , defined in equation (2.3), as the first member H (α) 0 = H (α) of a hierarchy of Hamiltonians…”

“…As stressed in [15], this imposes not only that they are square integrable on the (finite or infinite) interval of definition (x 1 , x 2 ) of V (x), as for the conventional Schrödinger equation, but also that they ensure the Hermiticity of H (α) (or that of π). The latter condition translates into…”

“…On building on the previously noted equivalence between the presence of a PDM and a deformation of the commutation relations [7], we have recently devised a method for generating pairs of potential and mass for which the Schrödinger equation is exactly solvable [15]. For such a purpose, we have used an approach inspired by a branch of supersymmetric quantum mechanics [17,18,19], whose development dates back to that of quantum groups and q-algebras [20,21,22,23,24,25,26].…”

“…It will then be a simple task to employ the reciprocal PCT to derive the PDM Schrödinger equation wavefunctions from the knowledge of the constant-mass ones. As we plan to show, this combined method works for all pairs of potential and mass considered in [15], including those already solved in [16,27]. This paper is organized as follows.…”