Exactly solvable quantum Sturm–Liouville problems

Abstract

The harmonic oscillator with time-dependent parameters covers a broad spectrum of physical problems from quantum transport, quantum optics, and quantum information to cosmology. Several methods have been developed to quantize this fundamental system, such as the path integral method, the Lewis–Riesenfeld time invariant method, the evolution operator or dynamical symmetry method, etc. In all these methods, solution of the quantum problem is given in terms of the classical one. However, only few exactly solvable problems of the last one, such as the damped oscillator or the Caldirola–Kanai model, have been treated. The goal of the present paper is to introduce a wide class of exactly solvable quantum models in terms of the Sturm–Liouville problem for classical orthogonal polynomials. This allows us to solve exactly the corresponding quantum parametric oscillators with specific damping and frequency dependence, which can be considered as quantum Sturm–Liouville problems.