00-213 Jonathan Butler
Semi-classical asymptotics of the spectral function of pseudodifferential operators (157K, AMS-TeX) May 8, 00
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Abstract. We consider the asymptotic behaviour of the spectral function of a self-adjoint $h$ pseudodifferential operator in the limit as $h \to 0$. Adapting methods developed in {\it The asymptotic distribution of eigenvalues of partial differential operators} by Yu. Safarov and D. Vassiliev to the semi-classical (non-homogeneous) setting, conditions are found under which a two-term asymptotic formula for the spectral function at a point on the diagonal may be written down, or under which so-called clustering of the spectral function occurs. To illustrate the results we consider the example of a Schr\"odinger operator $- h^2 \Delta + V$ with quadratic potential $V$.

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