Szegő limit theorems for the harmonic oscillator

Abstract:

Let $H = - \frac {1}{2}{d^2}/d{x^2} + \frac {1}{2}{x^2}$ be the harmonic oscillator Hamiltonian on ${L^2}( {\mathbf {R}})$, and let $A$ be a selfadjoint $DO$ of order $O$ in the Beals-Fefferman class with weights $\varphi = 1,\Phi (x,\xi ) = {(1 + |\xi {|^2} + |x {|^2})^{1/2}}$. Form the measure $\mu (f) = {\lim _{\lambda \to \infty }}(1/{\text {rank}}\;{\pi _\lambda }) {\text {tr}} f({\pi _\lambda } A{\pi _\lambda })$ where ${\pi _\lambda } A{\pi _\lambda }$ is the compression of $A$ onto the span of the Hermite functions with eigenvalue less than or equal to $\lambda$. Then one has the following Szegö limit theorem: \[ \mu (f) = \lim \limits _{T \to \infty } \;\frac {1} {{2 \pi T}}\;\int _{H(x,\xi ) \leqslant T} {f(a(x,\xi ))\;dx} \;d\xi \qquad {\text {for}}\ f \in C({\mathbf {R}}).\] For the special case where $f(x) = x$, this will be proved for a considerably wider class of operators by employing the Weyl correspondence. Moreover, by using estimates on Wigner functions of Hermite functions we are able to prove the full Szegö theorem for a fairly general class of multiplication operators.