Sharp Spectral Asymptotics for Operators with Irregular Coefficients. I. Pushing the Limits

Abstract

We consider operators on compact closed manifolds, with coefficients, first derivatives of which are continuous with continuity modulus and derive semiclassical spectral asymptotics with sharp remainder estimate O(h1−d); for operators with continuity modulus we derive semiclassical spectral asymptotics with the remainder estimate o(h1−d) under standard condition to Hamiltonian flow. For operators with even less regular coefficients we establish less sharp spectral asymptotics. These asymptotics easily yield standard asymptotics with respect to spectral parameter . We will treat operators on manifolds with boundary in the next paper.