Calculation of the propagator of Schr\"odinger's equation on $(0,\infty)$ with the potential $kx^{-2} + \omega^2x^2$ by Lie symmetry group method

TL;DR

This paper demonstrates how to derive the propagator for Schrödinger's equation with a combined harmonic oscillator and inverse-square potential on the half-line using Lie symmetry group methods, offering an alternative to Laplace's method.

Contribution

It introduces a symmetry group approach to compute the propagator for a specific quantum system, providing a new analytical method.

Findings

Propagator derived via Lie symmetry methods

Alternative to Laplace's method for the same problem

Enhances understanding of symmetry applications in quantum mechanics

Abstract

The propagators (fundamental solutions) of the heat and Schr\"odinger's equations on the half-line with a combined harmonic oscillator and inverse-square potential calculated in the recent paper [{\em J. Math. Phys.} {\bf 59}, 051507 (2018)] using Laplace's method are demonstrated to be obtainable alternatively within the framework of symmetry group methods discussed in a series of two papers in the same journal.