Propagators of isochronous an-harmonic oscillators and Mehler formula for the x-Hermite polynomials

TL;DR

This paper derives explicit elementary solutions for propagators of rationally extended harmonic oscillators and generalizes Mehler's formula to exceptional Hermite polynomials, providing computational algorithms and compact expressions.

Contribution

It introduces an algorithm for calculating propagators of extended oscillators and generalizes Mehler's formula to exceptional Hermite polynomials.

Findings

Explicit elementary solutions for propagators of extended oscillators.

Compact expressions for specific cases of propagators.

Generalized Mehler's formula for exceptional Hermite polynomials.

Abstract

It is shown that fundamental solutions $K^\sigma(x,y;t)=\langle x|e^{-i H^\sigma t}|y\rangle$ of the non-stationary Schr\"{o}dinger equation (Green functions, or propagators) for the rational extensions of the Harmonic oscillator $H^\sigma=H_{\rm osc}+\Delta V^\sigma$ are expressed in terms of elementary functions only. An algorithm to calculate explicitly $K_\sigma$ for an arbitrary $\sigma\in \mathbb{N}$ is given, compact expressions for $K^{\{1,2\}}$ and $K^{\{2,3\}}$ are presented. A generalization of the Mehler's formula to the case of exceptional Hermite polynomials is given.