Asymptotic expansions and stationary phase for operators with discontinuous symbol

Abstract

We consider the problem of the asymptotics of tr f(T_α) where f is a “general” function, T_α, the pseudodifferential operator with symbol σ(x,ξ,α). Here α is a large parameter, and σ is a rapidly decreasing function. If σ smooth, techniques which are by now standard yield complete asymptotic expansions, with computable coefficients, as α→∞. Here we consider also cases where σ is not smooth; it may jump across a hypersurface in x-space or a hyperplane in x, ξ-space. We prove the existence of complete asymptotic expansions and show how to compute the coefficients for polynomial f. Of central importance is the analytic continuations of certain distributions acting on spaces of non-smooth test functions.