Gaussian creation and annihilation operators ($a^{-}=D,a^{+}=x-D$)

On polynomials in $L^2$ of standard Gaussian measure, Gaussian integration by parts makes $D$ and $x-D$ adjoints, with commutator $[a^{-},a^{+}]=I$. The Probabilists' Hermite polynomials are obtained by applying $a^{+}$ repeatedly to $1$, and $a^{-}{\mathrm{He}}_n=n{\mathrm{He}}_{n-1}$. These give a Gaussian-space realization of creation and annihilation operators.

The normalized Probabilists' Hermite polynomials diagonalize the Gaussian number operator with eigenvalues $n\ge 0$. 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.

The energy form of the Gaussian number operator is the displayed squared derivative integral. Its form domain requires $\sum n|c_n{|}^2<\infty$, while the full operator domain requires $\sum n^2|c_n{|}^2<\infty$. It can be defined by ${\lim }_{t\downarrow 0}{t}^{-1}(\Vert f{\Vert }_2^2-⟨f,P_{t}f⟩)$ even when $Nf$ does not exist in $L^2$. This energy occurs in the Gaussian Poincaré inequality and Gaussian logarithmic Sobolev inequality.