Exactly Solvable Potentials and Quantum Algebras

TL;DR

This paper introduces a class of exactly solvable quantum potentials linked to quantum algebras, unifying several known potentials and revealing their algebraic structures and spectra.

Contribution

It presents a new finite-difference-differential equation framework that encompasses known potentials and connects to quantum algebra representations.

Findings

Potential solutions include Rosen-Morse, harmonic, and P"oschl-Teller potentials.

The algebraic structure involves $q$-deformed harmonic oscillator and $su_q(1,1)$.

Energy spectrum is exponential, with states forming reducible quantum algebra representations.

Abstract

A set of exactly solvable one-dimensional quantum mechanical potentials is described. It is defined by a finite-difference-differential equation generating in the limiting cases the Rosen-Morse, harmonic, and P\"oschl-Teller potentials. General solution includes Shabat's infinite number soliton system and leads to raising and lowering operators satisfying $q$-deformed harmonic oscillator algebra. In the latter case energy spectrum is purely exponential and physical states form a reducible representation of the quantum conformal algebra $su_q(1,1)$.