Mehler kernel

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In mathematics and physics, the Mehler kernel is the fundamental solution, (Green's function), or propagator of the Hamiltonian for the quantum harmonic oscillator. Mehler (1866) gave an explicit formula for it, called Mehler's formula.

The Mehler kernel provides the fundamental solution for the most general solution φ(xy; t) to

\frac{\partial \varphi}{\partial t} - \frac{\partial^2 \varphi}{\partial x^2}+x^2\varphi =0   ~.

Specifically, Mehler's kernel is

K(x,y;t)= \frac{\exp(-\coth(2t)(x^2+y^2)/2 - \text{cosech}(2t)xy)}{\sqrt{2\pi\sinh(2t)}}~.

By a simple transformation, this is, apart from a multiplying factor, the bivariate Gaussian probability density given by

\frac 1{2\pi \sqrt{1-\rho^2}}\exp\left(\frac{(x^2+y^2)- 2\rho xy}{1-\rho^2}\right)~.

When t = 0, variables x and y coincide, resulting in the limiting formula

K(x,y;0)= \delta(x-y)~.

The bivariate probability density can further be written as an infinite series involving the one-dimensional probability densities and Hermite polynomials of x and y.

The Kibble–Slepian formula generalizes this expansion to higher dimensions.