A new exactly solvable quantum model in N dimensions

TL;DR

This paper introduces an exactly solvable N-dimensional quantum model with position-dependent mass, curved kinetic term, and oscillator potential, which generalizes the isotropic oscillator to hyperbolic space with explicit spectra and eigenfunctions.

Contribution

It presents a new maximally superintegrable mbda-deformation of the N-dimensional oscillator with explicit solutions, extending solvable models to curved hyperbolic spaces.

Findings

Spectrum is hydrogenlike with explicit eigenvalues.

Eigenfunctions are obtained by deforming harmonic oscillator symmetry.

Model is exactly solvable for all positive mbda values.

Abstract

An N-dimensional position-dependent mass Hamiltonian (depending on a parameter \lambda) formed by a curved kinetic term and an intrinsic oscillator potential is considered. It is shown that such a Hamiltonian is exactly solvable for any real positive value of the parameter \lambda. Algebraically, this Hamiltonian can be thought of as a new maximally superintegrable \lambda-deformation of the N-dimensional isotropic oscillator and, from a geometric viewpoint, this system is just the intrinsic oscillator potential on an N-dimensional hyperbolic space with nonconstant curvature. The spectrum of this model is shown to be hydrogenlike, and their eigenvalues and eigenfunctions are explicitly obtained by deforming appropriately the symmetry properties of the N-dimensional harmonic oscillator. A further generalization of this construction giving rise to new exactly solvable models is envisaged.