Solution of the Cauchy Problem for a Time-Dependent Schoedinger Equation

TL;DR

This paper derives explicit solutions and propagators for a time-dependent Schrödinger equation with modified oscillator potentials, utilizing advanced mathematical functions and symmetries, and extends results to relativistic models.

Contribution

It introduces a novel explicit solution method for the time-dependent Schrödinger equation with a modified oscillator, incorporating symmetry and special functions, and generalizes to relativistic cases.

Findings

Explicit Green function for the modified oscillator

Solutions for relativistic oscillator models

Expansion formula for plane waves

Abstract

We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schroedinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU(1,1) symmetry of the harmonic oscillator wave functions, Bargmann's functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner-Pollaczek polynomials, a Hankel-type integral transform and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as a by-product.