Operator Theory: Advances and Applications

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2 Operator Theory: Advances and Applications Volume 214 Founded in 1979 by Israel Gohberg Editors: Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan (Odessa, Ukraine) Albrecht Böttcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa, IA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Lund, Sweden) Leonid E. Lerer (Haifa, Israel) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Leiba Rodman (Williamsburg, VA, USA) Ilya M. Spitkovsky (Williamsburg, VA, USA) Honorary and Advisory Editorial Board: Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J.William Helton (San Diego, CA, USA) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, USA) Subseries Linear Operators and Linear Systems Subseries editors: Daniel Alpay (Beer Sheva, Israel) Birgit Jacob (Wuppertal, Germany) André C.M. Ran (Amsterdam, The Netherlands) Subseries Advances in Partial Differential Equations Subseries editors: Bert-Wolfgang Schulze (Potsdam, Germany) Michael Demuth (Clausthal, Germany) Jerome A. Goldstein (Memphis, TN, USA) Nobuyuki Tose (Yokohama, Japan) Ingo Witt (Göttingen, Germany)

3 Spectral Theory and Analysis Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2008, Bedlewo, Poland Jan Janas Pavel Kurasov Ari Laptev Sergei Naboko Günter Stolz Editors

4 Editors Jan Janas Institut of Mathematics Polish Academy of Sciences ul. Sw. Tomasza 30/ Krakow, Poland Ari Laptev Department of Mathematics South Kensington Campus Imperial College London London SW7 2AZ, UK and Department of Mathematics Royal Institute of Technology Stockholm, Sweden Pavel Kurasov Department of Mathematics Lund Institute of Technology Lund, Sweden Sergei Naboko Department of Mathematical Physics St. Petersburg State University Ulyanovskaya St. Petersburg, Russia naboko@snoopy.phys.spbu.ru Günter Stolz Department of Mathematics University of Alabama 452 Campbell Hall Birmingham, AL , USA stolz@math.uab.edu 2010 Mathematical Subject Classification: Primary: (81Q10, 81Q12, 81Q15, 81Q20, 81Q80); Secondary: (35P05, 35P15, 35P20, 35P30), (47B25, 47F05) ISBN e-isbn DOI / Library of Congress Control Number: Springer Basel AG 2011 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, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media

5 This volume is dedicated to the memory and achievements of Mikhail Shlemovich Birman an outstanding mathematician of the 20th century in Mathematical Physics

6 Mikhail Shlemovich Birman

7 Contents Introduction... ix B.M. Brown, V. Hoang, M. Plum and I.G. Wood Floquet-Bloch Theory for Elliptic Problems with Discontinuous Coefficients... 1 D.E. Edmunds, W.D. Evans and D.J. Harris Representations of Compact Linear Operators in Banach Spaces and Nonlinear Eigenvalue Problems II R.L. Frank, A. Laptev and R. Seiringer A Sharp Bound on Eigenvalues of Schrödinger Operators on the Half-line with Complex-valued Potentials Yu.E. Karpeshina Zero-range Model of p-scattering by a Potential Well A.V. Kiselev The Similarity Problem for Non-selfadjoint Operators with Absolutely Continuous Spectrum: Restrictions to Spectral Subspaces D.Lenz,P.StollmannandI.Veselić Generalized Eigenfunctions and Spectral Theory for Strongly Local Dirichlet Forms A. Mikikits-Leitner and G. Teschl Trace Formulas for Schrödinger Operators in Connection with Scattering Theory for Finite-gap Backgrounds A. Tikhonov Inner-outer Factorization for Weighted Schur Class Functions and Corresponding Invariant Subspaces F. Truc Eigenvalue Asymptotics for Magnetic Fields and Degenerate Potentials List of Participants

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9 Introduction This volume appears as an outcome of the conference Operator Theory, Analysis and Mathematical Physics OTAMP2008, held at Mathematical Research and Conference Center in Bedlewo near Poznan. The volume contains few review articles as well as original research papers presented at the conference or appeared as a result of inspiring discussions during the meeting. The titles of conference talks followed the subjects of four special sessions: random and quasi-periodic differential operators orthogonal polynomials Jacobi and CMV matrices quantum graphs All contributions to this volume are devoted to different chapters of operator theory with a focus towards applications in mathematical physics. Several articles are in the area of spectral theory of Schrödinger operators having emphasis on problems with magnetic fields. Another subject well-represented concerns spectral theory for non-self-adjoint problems. Spectral analysis is not restricted to just linear and self-adjoint problems. This volume is devoted to the memory of Mikhail Shlemovich Birman one of the most outstanding scientists of the last century. The influence of his ideas on the development of mathematical physics in the whole world and especially in Saint Petersburg will continue for decades, several of the authors contributed to this volume have been his students and will carry over his special attitude towards science to new generations to come. Preparing this volume we remembered another remarkable mathematical physicist Israel Gohberg who always supported OTAMP conferences by including proceedings into the series Operator Theory: Advances and Applications and helping us with selection of outstanding contributors and interesting subjects in operator theory. We would like to thank the European Science Foundation (ESF) for a generous financial support which allowed to transfer the OTAMP conference into a major event in the area of mathematical physics in We are grateful to all people working at Mathematical Research and Conference Center in Bedlewo for creating a stimulating scientific atmosphere and help before, during and after the conference. Birmingham-Krakow-London Lund-St. Petersburg-Stockholm May 2010 The Editors