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Transactions of the American Mathematical Society

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Szegő limit theorems for the harmonic oscillator

Authors: A. J. E. M. Janssen and Steven Zelditch
Journal: Trans. Amer. Math. Soc. 280 (1983), 563-587
MSC: Primary 35S05; Secondary 81C10
MathSciNet review: 716838
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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 + \vert\xi \,{\vert^2} + \vert x\,{\vert^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:

$\displaystyle \mu (f) = \mathop {\lim }\limits_{T \to \infty } \;\frac{1} {{2\,... ...qslant T} {f(a(x,\xi ))\;dx} \;d\xi \qquad {\text{for}}\ f \in C({\mathbf{R}}).$

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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