Gaussian number operator
= Gaussian number operator
{c}
{title2=$N=a^+a^-=-D^2+xD$}
The normalized <Probabilists' Hermite polynomials> diagonalize the Gaussian number operator with <eigenvalues> $n\geq0$. Its self-adjoint closure has domain $\sum n^2|c_n|^2<\infty$. Polynomial truncations are dense for its graph <norm>. The semigroup $e^{-tN}$ is the <Ornstein-Uhlenbeck semigroup> and its energy form is the <Gaussian Dirichlet energy>.