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. 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. The NATO Advanced Study Institute "Microlocal Analysis and Spectral TheƯ 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 equaƯ 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 linƯ 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 MathƯ ematical Analysis, with other applications all around Mathematics and Physics, one major theme being Spectral Theory for Schrodinger equation in Quantum Mechanics Front Matter....Pages i-viii Linear Partial Differential Equations with Multiple Involutive Characteristics....Pages 1-38 Gevrey and Analytic Hypoellipticity....Pages 39-59 Higher Microlocalization and Propagation of Singularities....Pages 61-90 Conormality and Lagrangian Properties in Diffractive Boundary Value Problems....Pages 91-113 Parametrized Pseudodifferential Operators and Geometric Invariants....Pages 115-164 Boundary Value Problems and Edge Pseudo-Differential Operators....Pages 165-226 Wodzicki’s Noncommutative Residue and Traces for Operator Algebras on Manifolds with Conicai Singularities....Pages 227-250 Lower Bounds for Pseudodifferential Operators....Pages 251-262 Weyl Formula for Globally Hypoelliptic Operators in R n ....Pages 263-306 Splitting in Large Dimension and Infrared Estimates....Pages 307-347 Microlocal Exponential Estimates and Applications to Tunneling....Pages 349-376 A Trace Formula and Review of Some Estimates for Resonances....Pages 377-437 Back Matter....Pages 439-444