Sharp Spectral Asymptotics for Operators with Irregular Coefficients. II. Domains with Boundaries and Degenerations
Victor Yakovlevich Ivrii · Communications in Partial Differential Equations · 2003
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.