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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