We show that there exist some intimate connections between three unconventional Schrodinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space. This occurs whenever a specific relation between the deforming function, the position-dependent mass and the (diagonal) metric tensor holds true. We illustrate these three equivalent approaches by considering a new Coulomb problem and solving it by means of supersymmetric quantum mechanical and shape invariance techniques. We show that in contrast with the conventional Coulomb problem, the new one gives rise to only a finite number of bound states.

Deformed algebras, position-dependent effective masses and curved spaces: an exactly solvable Coulomb problem

Known shape-invariant potentials for the constant-mass Schrodinger 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.

https://iopscience.iop.org/article/10.1088/0305-4470/38/13/008/pdf

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

Series solutions of the Schrödinger equation with position-dependent mass for the Morse potential

We show that there exist some intimate connections between three unconventional Schr\"odinger equations based on the use of deformed canonical commutation relations, of a position-dependent effective mass or of a curved space, respectively. This occurs whenever a specific relation between the deforming function, the position-dependent mass and the (diagonal) metric tensor holds true. We illustrate these three equivalent approaches by considering a new Coulomb problem and solving it by means of supersymmetric quantum mechanical and shape invariance techniques. We show that in contrast with the conventional Coulomb problem, the new one gives rise to only a finite number of bound states.

Deformed algebras, position-dependent effective masses and curved spaces: An exactly solvable Coulomb problem

Exactly solvable potentials for the Schrödinger equation with spatially dependent mass

We discuss the relationship between exact solvability of the Schrodinger equation with a position-dependent mass and the ordering ambiguity in the Hamiltonian operator within the framework of supersymmetric quantum mechanics. The one-dimensional Schrodinger equation, derived from the general form of the effective mass Hamiltonian, is solved exactly for a system with exponentially changing mass in the presence of a potential with similar behaviour, and the corresponding supersymmetric partner Hamiltonians are related to the effective-mass Hamiltonians proposed in the literature.

Supersymmetric approach to exactly solvable systems with position-dependent effective masses

We discuss the relationship between exact solvability of the Schr\"{o}dinger equation with a position-dependent mass and the ordering ambiguity in the Hamiltonian operator within the frame of supersymmetric quantum mechanics. The one-dimensional Schr\"{o}dinger equation, derived from the general form of the effective mass Hamiltonian, is solved exactly for a system with exponentially changing mass in the presence of a potential with similar behaviour, and the corresponding supersymmetric partner Hamiltonians are related to the effective-mass Hamiltonians proposed in the literature.

D. Dot Ilek Tutcu

Papers

Supersymmetric approach to exactly solvable systems with position-dependent effective masses

Supersymmetric approach to exactly solvable systems with position-dependent effective masses