Noncommutative Microlocal Analysis

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.

This book explores some basic roles of Lie groups in linear analysis, with particular emphasis on the generalizations of the Fourier transform and the study of partial differential equations. It began as lecture notes for a one-semester graduate course given by the author in noncommutative harmonic analysis. It is a valuable resource for both graduate students and faculty, and requires only a background with Fourier analysis and basic functional analysis, plus the first few chapters of a standard text on Lie groups. The basic method of noncommutative harmonic analysis, a generalization of Fourier analysis, is to synthesize operators on a space on which a Lie group has a unitary representation from operators on irreducible representation spaces.Though the general study is far from complete, this book covers a great deal of the progress that has been made on important classes of Lie groups. Unlike many other books on harmonic analysis, this book focuses on the relationship between harmonic analysis and partial differential equations. The author considers many classical PDEs, particularly boundary value problems for domains with simple shapes, that exhibit noncommutative groups of symmetries. Also, the book contains detailed work, which has not previously been published, on the harmonic analysis of the Heisenberg group and harmonic analysis on cones.

Microlocal analysis is a field of mathematics that was invented in the mid-20th century for the detailed investigation of problems from partial differential equations, which incorporated and made rigorous many ideas that originated in physics. Since then it has grown to a powerful machine which is used in global analysis, spectral theory, mathematical physics and other fields, and its further development is a lively area of current mathematical research. In this book extended abstracts of the conference 'Microlocal Methods in Mathematical Physics and Global Analysis', which was held at the University of Tübingen from the 14th to the 18th of June 2011, are collected.​

This volume is the proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Microlocal Analysis and its Applications to Partial Differential Equations, held July 10-16, 1983 in Boulder, Colorado. It contains refereed articles which were delivered at the conference. Two of the papers are survey articles, one on uniqueness and non-uniqueness in the Cauchy problem and one on hypoanalytic structures; the rest are either detailed announcements or complete papers covering such areas as spectrum of operators, nonlinear problems, asymptotics, pseudodifferential operators of multiple characteristics and operators on groups and homogeneous spaces. The volume should be useful to active mathematicians and graduate students working on linear and nonlinear partial differential equations and related areas.

This book provides the first coherent account of the area of analysis that involves the Heisenberg group, quantization, the Weyl calculus, the metaplectic representation, wave packets, and related concepts. This circle of ideas comes principally from mathematical physics, partial differential equations, and Fourier analysis, and it illuminates all these subjects. The principal features of the book are as follows: a thorough treatment of the representations of the Heisenberg group, their associated integral transforms, and the metaplectic representation; an exposition of the Weyl calculus of pseudodifferential operators, with emphasis on ideas coming from harmonic analysis and physics; a discussion of wave packet transforms and their applications; and a new development of Howe's theory of the oscillator semigroup.

This book grew out of a series of lectures given at the Mathematics Department of Kyushu University in the Fall 2006, within the support of the 21st Century COE Program (2003–2007) “Development of Dynamical Mathematics with High Fu- tionality” (Program Leader: prof. Mitsuhiro Nakao). It was initially published as the Kyushu University COE Lecture Note n- ber 8 (COE Lecture Note, 8. Kyushu University, The 21st Century COE Program “DMHF”, Fukuoka, 2008. vi+234 pp.), and in the present form is an extended v- sion of it (in particular, I have added a section dedicated to the Maslov index). The book is intended as a rapid (though not so straightforward) pseudodiff- ential introduction to the spectral theory of certain systems, mainly of the form a +a where the entries of a are homogeneous polynomials of degree 2 in the 2 0 2 n n (x,?)-variables, (x,?)? R×R,and a is a constant matrix, the so-called non- 0 commutative harmonic oscillators, with particular emphasis on a class of systems introduced by M. Wakayama and myself about ten years ago. The class of n- commutative harmonic oscillators is very rich, and many problems are still open, and worth of being pursued.