Exactly solvable potentials and quantum algebras

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. A general solution includes Shabat's infinite number soliton system and leads to raising and lowering operators satisfying a q-deformed harmonic-oscillator algebra. In the latter case the energy spectrum is purely exponential and physical states form a reducible representation of the quantum conformal algebra ${\mathrm{su}}_{\mathit{q}}$(1,1).