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

It is shown that fundamental solutions Kσ(x,y;t)=〈x|e −i Hσt|y〉 of the non-stationary Schrödinger equation (Green functions, or propagators) for the rational extensions of the Harmonic oscillator Hσ=Hosc+ΔVσ are expressed in terms of elementary functions only. An algorithm to calculate explicitly Kσ for an arbitrary increasing sequence of positive integers σ is given, and compact expressions for K{1,2} and K{2,3} are presented. A generalization of Mehler's formula to the case of exceptional Hermite polynomials is given.