Spectral Theory of Infinite-Area Hyperbolic Surfaces
This book introduces geometric spectral theory in the context of infinite-area Riemann surfaces, providing a comprehensive account of dramatic recent developments in the field. These developments were prompted by advances in geometric scattering theory in
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Series Editors Hyman Bass Joseph Oesterl´e Alan Weinstein
David Borthwick
Spectral Theory of Infinite-Area Hyperbolic Surfaces
Birkh¨auser Boston • Basel • Berlin
David Borthwick Department of Mathematics and Computer Science Emory University Atlanta, GA 30322 U.S.A. [email protected]
Mathematics Subject Classification (2000): 58J50, 47A40, 11F72, 30F35 Library of Congress Control Number: 2007932363 ISBN-13: 978-0-8176-4524-3
e-ISBN-13: 978-0-8176-4653-0
Printed on acid-free paper. c 2007 Birkh¨auser Boston
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkh¨auser Boston, c/o Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 987654321 www.birkhauser.com
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For Sarah, Julia, and Benjamin
Preface
I first encountered the spectral theory of hyperbolic surfaces as an undergraduate physics student, through the intriguing expository article of Balazs–Voros [11] on relations between the Selberg theory of automorphic forms and quantum chaos. At the time I was quite impressed at the range of topics represented, including quantum physics, discrete groups, differential geometry, number theory, complex analysis, and spectral theory. In my previous experience these were completely separate realms, but here they were all mixed together in the same setting. Twenty years later, these topics do not seem so far apart to me. However, I am no less amazed by the rich cross-fertilization of ideas in this subject area. The primary motivation for this book is the conviction that this sort of mathematics that bridges the divides between fields ought to be made accessible to as broad an audience as possible—to graduate students especially, for whom regular coursework often exaggerates the impression of boundaries between disciplines. The spectral theory of compact and finite-area Riemann surfaces is a classical subject with a history going back to the pioneering work of Atle Selberg, who brought techniques from spectral theory and harmonic analysis into the study of automorphic forms. These cases have been thoroughly covered in various expository sources. In particular, Buser [37] develops the spectral theory for compact Riemann surfaces with a concrete approach based on hyperbolic geometry and cutting and pasting. Most treatments of the finite-area case, for example Venkov [210], emphasize arithmetic surfaces and connections to number theory. For infinite-area hyperbolic surfaces, a good unde
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