Sharp spectral asymptotics for operators with irregular coefficients. II. Domains with boundaries and degenerations

This paper is a continuation of another (Bronstein M, Ivrii V. Sharp spectral asymptotics for operators with irregular coefficients. I. Pushing the limits. Commun Part Diff Equal 2003; 28(1&2):99-123) in which we derived spectral asymptotics with sharp remainder estimates for operators on compact closed manifolds, with coefficients, first derivatives of which are continuous with continuity modulus O(\log\x - y\\(-1)). Now we derive semiclassical spectral asymptotics with the same sharp remainder estimate O(h(1-d)) for operators on manifolds with the boundary which also satisfies very minimal regularity condition. We also derive semiclassical spectral asymptotics with the remainder estimate o(h(l-d)) under standard condition to Hamiltonian flow: the sets of dead-end and periodic points both have measure zero. Moreover, we get rid of or relax microhyperbolicity condition for scalar operators.