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

Abstract

It is shown that fundamental solutions $K^{\sigma }(x,y;t)=⟨x|e^{-i{H}^{\sigma }t}|y⟩$ of the non-stationary Schrödinger equation (Green functions, or propagators) for the rational extensions of the Harmonic oscillator $H^{\sigma }=H_{\mathrm{osc}}+\Delta V^{\sigma }$ 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.

References in corpus (1)

• Exact propagators for SUSY partners