Microlocal Analysis, Sharp Spectral Asymptotics and Applications I: Semiclassical Microlocal Analysis and Local and Microlocal Semiclassical Asymptotics

Mathematics Subject Classification (2010): • 35P20 Asymptotic distributions of eigenvalues in context of PDEs • 35S05 Pseudodifferential operators as generalizations of partial differential operators • 35S30 Fourier integral operators applied to PDEs • 81V70 Many-body theory; quantum Hall effect

The prime goal of this monograph, which comprises a total of five volumes, is to derive sharp spectral asymptotics for broad classes of partial differential operators using techniques from semiclassical microlocal analysis, in particular, propagation of singularities, and to subsequently use the variational estimates in “small” domains to consider domains with singularities of different kinds. In turn, the general theory (results and methods developed) is applied to the Magnetic Schrödinger operator, miscellaneous problems, and multiparticle quantum theory.

In this volume the general microlocal semiclassical approach is developed, and microlocal and local semiclassical spectral asymptotics are derived.