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 Equat 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 . Now we derive semiclassical spectral asymptotics with the same sharp remainder estimateO(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. This paper is a continuation of another [1] in which semiclassical spectral asymptotics with remainder estimate ' were proven for operators on the closed manifolds with the first derivatives of the coefficients continuous with the continuity modulus ; for under standard condition to the Hamiltonian flow remainder estimate o(h 1-d ) was achieved. Apart of the lack of the boundary, the microhyperbolicity condition was assumed; for a scalar operator this condition means that as a τ where a is the symbol of operator and τ is the energy level in question. In this paper by means of new arguments (partially close to those of [4] ) we extend these results to In the next papers (5 etc) we will study Schrödinger and Dirac operators with strong magnetic field and then continue our main project: to recover Book (i.e. [3] ) under these weak regularity conditions.