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

英文摘要：

It is shown that fundamental solutions $K^σ(x,y;t)=\langle x|e^{-i H^σt}|y\rangle$ of the non-stationary Schrödinger equation (Green functions, or propagators) for the rational extensions of the Harmonic oscillator $H^σ=H_{\rm osc}+ΔV^σ$ are expressed in terms of elementary functions only. An algorithm to calculate explicitly $K_σ$ for an arbitrary $σ\in \mathbb{N}$ 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.