Jonathan Butler
Semi-classical asymptotics of the spectral function of pseudodifferential 
operators
(157K, AMS-TeX)

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 $.