Intermediate Spectral Theory and Quantum Dynamics
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Editors-in-Chief Anne Boutet de Monvel (Université Paris VII Denis Diderot, France) Gerald Kaiser (Center for Signals and Waves, Austin, TX, USA)
Editorial Board C. Berenstein (University of Maryland, College Park, USA) Sir M. Berry (University of Bristol, UK) P. Blanchard (University of Bielefeld, Germany) M. Eastwood (University of Adelaide, Australia) A.S. Fokas (University of Cambridge, UK) D. Sternheimer (Université de Bourgogne, Dijon, France) C. Tracy (University of California, Davis, USA)
César R. de Oliveira
Intermediate Spectral Theory and Quantum Dynamics
Birkhäuser Basel · Boston · Berlin
Author: César R. de Oliveira Department of Mathematics Federal University of São Carlos (UFSCar) São Carlos, SP 13560-970 Brazil e-mail: [email protected]
2000 Mathematics Subject Classification: 00-01, 81Q10 , 47A05 , 47A07, 47B25, 42B10
Library of Congress Control Number: 2008935753
Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de
ISBN 978-3-7643-8794-5 Birkhäuser Verlag AG, Basel · Boston · Berlin 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, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2009 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-7643-8794-5 987654321
e-ISBN 978-3-7643-8795-2 www.birkhauser.ch
Dedicated to my parents (Marly, Jo˜ao) and my parents-in-law (Elza, Geraldo)
Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Selected Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
A Glance at Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . .
1
1 Linear Operators and Spectra 1.1 Bounded Operators . . . . . . . 1.2 Closed Operators . . . . . . . . 1.3 Compact Operators . . . . . . 1.4 Hilbert-Schmidt Operators . . . 1.5 The spectrum . . . . . . . . . . 1.6 Spectra of Compact Operators
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5 15 20 27 32 39
2 Adjoint Operator 2.1 Adjoint Operator . . . . . . . . 2.2 Cayley Transform I . . . . . . . 2.3 Examples . . . . . . . . . . . . 2.3.1 Momentum and Energy 2.3.2 Multiplication Operator 2.4 Weyl Sequences . . . . . . . . . 2.5 Cayley Transform II . . . . . . 2.6 Examples . . . . . . . . . . . .
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