........ published in NEWSLETTER # 63
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........ published in NEWSLETTER # 63
MICROLOCAL ANALYSIS AND SPECTRAL THEORY
By Professor L. Rodino, University, Turin (Italy)
The aim of this volume (NATO ASI SERIES C490) is to record the considerable progress made 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 equations, where now the localization proceeding takes place with respect to the dual variables as well. Through the tools of pseudo-differential operators, wace-front sets and Fourier integral operators, the general theory of the linear 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 into a definite and independent part of mathematical analysis, with other applications all around mathematics and physics, one major theme being spectral theory for Schrodinger equation in quantum mechanics.
Concerning the general theory of linear PDE, contributions are presented in the following directions: new topics in the Gevrey- analytic category, as propagation and hypoellipticity in the case of multiple characteristics; higher analytic microlocalization; advances in elliptic boundary value problems, in particular on manifolds with singularities.
Concerning spectral theory, different problems for the Schrodinger equation are discussed, in particular: asymptotic behaviour of the eigenvalues, semiclassical analysis in large dimension and statistical mechanics, tunneling and adiabatic theory, asymptotic of resonances.
As a predecessor of the present volume we quote "Advances in Microlocal Analysis" edited by G. Garnir (NATO ASI Series, D. Reidel Publishing Company 1986, Series C. Mathematical and Physical Sciences Vol. no. C 168). The progress in this area is so rapid, that further NATO activities in Microlocal Analysis are expected in a few years, addressed mainly to the new field of non- linear equations.
Reference books: C168, C490