Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction
This volume describes the spectral theory of the Weyl quantization of systems of polynomials in phase-space variables, modelled after the harmonic oscillator. The main technique used is pseudodifferential calculus, including global and semiclassical varia
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1992
Alberto Parmeggiani
Spectral Theory of Non-Commutative Harmonic Oscillators: An Introduction
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Alberto Parmeggiani Department of Mathematics University of Bologna Piazza di Porta San Donato, 5 40126 Bologna Italy [email protected]
ISBN: 978-3-642-11921-7 e-ISBN: 978-3-642-11922-4 DOI: 10.1007/978-3-642-11922-4 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010924182 Mathematics Subject Classification (2000): 35P20, 35J45, 35P15, 35S05, 47A10, 47G30, 47N50, 58J52, 81Q20 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
To Serena & Lorenzo, Luisa & Pier Luigi
Preface
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 Functionality” (Program Leader: prof. Mitsuhiro Nakao). It was initially published as the Kyushu University COE Lecture Note number 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 version 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) pseudodifferential introduction to the spectral theory of certain systems, mainly of the form a2 + a0 where the entries of a2 are homogeneous polynomials of degree 2 in the (x, ξ )-variables, (x, ξ ) ∈ Rn × Rn , and a0 is a constant matrix, the so-called noncommutative harmonic oscillators, with particular emphasis on a class of systems introduced by M. Wakayama and myself about ten years ago. The class of noncommutative harmonic oscillators is very rich, and many problems are still open, and worth of being pursued. I wish to thank Masato Wakayama, dearest friend and collaborator, and my friends and colleagues Nicola Arcozzi, Sandro Coriasco, Sandro Graffi, Fr´ed´eric H´erau, Takashi Ichinose, Miyuki Kuze, Lidia Maniccia, Luca Migliorini, Cesare Parenti and
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