# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

## Szegő limit theorems for the harmonic oscillatorHTML articles powered by AMS MathViewer

by A. J. E. M. Janssen and Steven Zelditch
Trans. Amer. Math. Soc. 280 (1983), 563-587 Request permission

## 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.
Similar Articles
• Retrieve articles in Transactions of the American Mathematical Society with MSC: 35S05, 81C10
• Retrieve articles in all journals with MSC: 35S05, 81C10