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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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Szegő limit theorems for the harmonic oscillator
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by A. J. E. M. Janssen and Steven Zelditch PDF
Trans. Amer. Math. Soc. 280 (1983), 563-587 Request permission


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.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 280 (1983), 563-587
  • MSC: Primary 35S05; Secondary 81C10
  • DOI:
  • MathSciNet review: 716838