Microlocal Analysis and Spectral Theory

There has been considerable recent progress in the field of microlocal analysis. In a broad sense the subject is the modern version of the classical Fourier technique for solving partial differential equations, with the localization process taking account of dual variables. The tools of pseudo-differential operators, wave-front sets and Fourier integral operators have now conferred a mature form on the theory of linear partial differential operators in the frame of Schwartz distributions or other generalized functions. At the same time, microlocal analysis has assumed an important role as an independent part of analysis, with other applications throughout mathematics and physics, one major theme being spectral theory for the Schrödinger equation in quantum mechanics.
The papers collected here emphasize the topics of microlocal methods in the study of linear PDEs (analytic-Gevrey regularity of the solutions, elliptic boundary value problems, higher microlocalization), and applications to spectral theory (Schrödinger equation, asymptotic behavior of the eigenvalues, semi-classical analysis in large dimensions and statistical mechanics).
Audience: Accessible to a wide audience, including graduate students in analysis and non-specialists from mathematics and physics.

Inhalt

Preface. Linear Partial Differential Equations with Multiple Involutive Characteristics; O. Liess, L. Rodino. Gevrey and Analytic Hypoellipticity; D.S. Tartakoff. Higher Microlocalization and Propagation of Singularities; O. Liess. Conormality and Lagrangian Properties in Diffractive Boundary Value Problems; P. Laubin. Parametrized Pseudodifferential Operators and Geometric Invariants; G. Grubb. Boundary Value Problems and Edge Pseudodifferential Operators; B.-W. Schulze. Wodzicki's Noncommutative Residue and Traces for Operator Algebras on Manifolds with Conical Singularities; E. Schrohe. Lower Bounds for Pseudodifferential Operators; C. Parenti, A. Parmeggiani. Weyl Formula for Globally Hypoelliptic Operator in Rn; E. Buzano. Splitting in Large Dimension and Infrared Estimates; B. Helffer. Microlocal Exponential Estimates and Applications to Tunneling; A. Martinez. A Trace Formula and Review of Some Estimates for Resonances; J. Sjöstrand. Index.