Microlocal Analysis and Spectral Theory

The NATO Advanced Study Institute "Microlocal Analysis and Spectral The­ ory" was held in Tuscany (Italy) at Castelvecchio Pascoli, in the district of Lucca, hosted by the international vacation center "11 Ciocco" , from September 23 to October 3, 1996. The Institute recorded the considerable progress realized recently in the field of Microlocal Analysis. In a broad sense, Microlocal Analysis is the modern version of the classical Fourier technique in solving partial differential equa­ tions, where now the localization proceeding takes place with respect to the dual variables too. Precisely, through the tools of pseudo-differential operators, wave-front sets and Fourier integral operators, the general theory of the lin­ ear partial differential equations is now reaching a mature form, in the frame of Schwartz distributions or other generalized functions. At the same time, Microlocal Analysis has grown up into a definite and independent part of Math­ ematical Analysis, with other applications all around Mathematics and Physics, one major theme being Spectral Theory for Schrodinger equation in Quantum Mechanics.

Linear Partial Differential Equations with Multiple Involutive Characteristics.- Gevrey and Analytic Hypoellipticity.- Higher Microlocalization and Propagation of Singularities.- Conormality and Lagrangian Properties in Diffractive Boundary Value Problems.- Parametrized Pseudodifferential Operators and Geometric Invariants.- Boundary Value Problems and Edge Pseudo-differential Operators.- Wodzicki’s Noncommutative Residue and Traces for Operator Algebras on Manifolds with Conical Singularities.- Lower Bounds for Pseudodifferential Operators.- Weyl Formula For Globally Hypoelliptic Operator in Rn.- Splitting in large dimension and infrared estimates.- Microlocal Exponential Estimates and Applications to Tunneling.- A trace formula and review of some estimates for resonances.