Mehler kernel

In mathematics, the Mehler kernel is the heat kernel of the Hamiltonian of the harmonic oscillator. Mehler (1866) gave an explicit formula for it called Mehler's formula. The Kibble–Slepian formula generalizes Mehler's formula to higher dimensions.

The Mehler kernel φ(x, y, t) is a solution to

\frac{\partial \varphi}{\partial t} = \frac{\partial^2 \varphi}{\partial x^2}-x^2\varphi

Mehler's kernel is

\frac{\exp(-\coth(2t)(x^2%2By^2)/2 - \text{cosech}(2t)xy)}{\sqrt{2\pi\sinh(2t)}}

This can also be written as an infinite series involving Hermite polynomials of x and y.

References