Semi-classical asymptotics of the spectral function of pseudodifferential operators

We consider the asymptotic behaviour of the spectral function of a self-adjoint h pseudodifferential operator in the limit as h --> 0. Adapting methods developed in (1) to the semiclassical (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 Schrodinger operator -h(2)Delta + V with quadratic potential V.