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

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.