 00213 Jonathan Butler
 Semiclassical asymptotics of the spectral function of pseudodifferential
operators
(157K, AMSTeX)
May 8, 00

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Abstract. We consider the asymptotic behaviour of the spectral function of a
selfadjoint $ 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 semiclassical (nonhomogeneous)
setting, conditions are found under which a twoterm asymptotic
formula for the spectral function at a point on the diagonal may be
written down, or under which socalled 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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