Contributors

Editor Ari Laptev

President The European Mathematical Society

Professor Head of Department Department of Mathematics Imperial College London Huxley Building, 180 Queen’s Gate London SW7 2AZ, UK [email protected]

Professor Department of Mathematics Royal Institute of Technology 100 44 Stockholm, Sweden [email protected]

Ari Laptev is a world-recognized specialist in Spectral Theory of Differential Operators. He discovered a number of sharp spectral and functional inequali- ties. In particular, jointly with his former student T. Weidl, A. Laptev proved sharp Lieb–Thirring inequalities for the negative spectrum of multidimen- sional Schr¨odingeroperators, a problem that was open for more than twenty five years. A. Laptev was brought up in Leningrad (Russia). In 1971, he graduated from the Leningrad State University and was appointed as a researcher and then as an Assistant Professor at the Mathematics and Mechanics Depart- ment of LSU. In 1982, he was dismissed from his position at LSU due to his marriage to a British subject. Only after his emigration from the USSR in 1987 he was able to continue his career as a mathematician. Then A. Laptev was employed in Sweden, first as a lecturer at Link¨opingUniversity and then from 1992 at the Royal Institute of Technology (KTH). In 1999, he became a professor at KTH and also Vice Chairman of its Department of Mathematics. From January 2007 he is employed by Imperial College London where from September 2008 he is the Head of Department of Mathematics. A. Laptev was the Chairman of the Steering Committee of the five years long ESF Programme SPECT, the President of the Swedish Mathematical Society from 2001 to 2003, and the President of the Organizing Committee of the Fourth European Congress of Mathematics in Stockholm in 2004. He is now the President of the European Mathematical Society for the period January 2007–December 2010. Authors

David R. Adams Vol. III Andrea Cianchi Vol. I University of Kentucky Universit`adi Firenze Lexington, KY 40506-0027 Piazza Ghiberti 27, 50122 Firenze USA ITALY [email protected] cianchi@unifi.it Hiroaki Aikawa Vol. III Martin Costabel Vol. I Hokkaido University Universit´ede Rennes 1 Sapporo 060-0810 Campus de Beaulieu JAPAN 35042 Rennes [email protected] FRANCE [email protected] Farit Avkhadiev Vol. I Kazan State University Monique Dauge Vols. I, II 420008 Kazan Universit´ede Rennes 1 RUSSIA Campus de Beaulieu [email protected] 35042 Rennes FRANCE Catherine Bandle Vol. II [email protected] Mathematisches Institut Universit¨atBasel Martin Dindoˇs Vol. II Rheinsprung 21, CH-4051 Basel Maxwell Institute of SWITZERLAND Mathematics Sciences [email protected] University of Edinburgh JCMB King’s buildings Mayfield Rd Gerassimos Barbatis Vol. II Edinbugh EH9 3JZ University of Athens UK 157 84 Athens [email protected] GREECE [email protected] Andr´asDomokos Vol. II California State Sergey Bobkov Vol. I University Sacramento University of Minnesota Sacramento 95819 Minneapolis, MN 55455 USA USA [email protected] [email protected] Yuri V. Egorov Vol. II Victor I. Burenkov Vol. II Universit´ePaul Sabatier Universit`adegli Studi di Padova 118 route de Narbonne 63 Via Trieste, 35121 Padova 31062 Toulouse Cedex 9 ITALY FRANCE [email protected] [email protected] Grigori Chechkin Vol. II Gregory Eskin Vol. III Lomonosov Moscow State University University of California Vorob’evy Gory, Moscow Los Angeles, CA 90095-1555 RUSSIA USA [email protected] [email protected] Alberto Cialdea Vol. III Nicola Garofalo Vol. I Universit`adella Basilicata Purdue University Viale dell’Ateneo Lucano 10, West Lafayette, IN 47906 85100, Potenza USA ITALY [email protected] [email protected] and Authors xi

Universit`adi Padova Niels Jacob Vol. I 35131 Padova Swansea University ITALY Singleton Park [email protected] Swansea SA2 8PP Vladimir Gol’dshtein Vol. I UK [email protected] Ben Gurion University of the Negev P.O.B. 653, Beer Sheva 84105 Dmitry Khavinson Vol. II ISRAEL University of South Florida [email protected] 4202 E. Fowler Avenue, PHY114 Alexander Grigor’yan Vol. II Tampa, FL 33620-5700 USA Bielefeld University Bielefeld 33501 [email protected] GERMANY Juha Kinnunen Vol. I [email protected] Institute of Mathematics Stathis Filippas Vol. I Helsinki University of Technology University of Crete P.O. Box 1100, FI-02015 71409 Heraklion FINLAND juha.kinnunen@tkk.fi Institute of Applied and Computational Mathematics Gerasim Kokarev Vol. II 71110 Heraklion University of Edinburgh GREECE King’s Buildings, Mayfield Road fi[email protected] Edinburgh EH9 3JZ Rupert L. Frank Vol. I UK [email protected] Princeton University Washington Road Vladimir A. Kondratiev Vol. II Princeton, NJ 08544 Moscow State University USA 119992 Moscow [email protected] RUSSIA Michael W. Frazier Vol. III [email protected] University of Tennessee Riikka Korte Vol. I Knoxville, Tennessee 37922 University of Helsinki USA P.O. Box 68 [email protected] Gustaf H¨allstr¨ominkatu 2 b Bernard Helffer Vol. III FI-00014 FINLAND Universit´eParis-Sud riikka.korte@helsinki.fi 91 405 Orsay Cedex FRANCE Pekka Koskela Vol. I Bernard.Helff[email protected] University of Jyv¨askyl¨a Thomas Hoffmann-Ostenhof Vol. III P.O. Box 35 (MaD), FIN–40014 FINLAND Institut f¨urTheoretische Chemie [email protected].fi Universit¨atWien W¨ahringerStrasse 17, and Nikolay Kuznetsov Vol. II International Erwin Institute for Problems Schr¨odingerInstitute in Mechanical Engineering for Mathematical Physics Russian Academy of Sciences Boltzmanngasse 9 V.O., Bol’shoy pr. 61 A-1090 Wien 199178 St. Petersburg AUSTRIA RUSSIA thoff[email protected] [email protected] Volodymyr Hrynkiv Vol. III Pier Domenico Lamberti Vol. II University of Houston-Downtown Universit´adegli Studi di Padova Houston, TX 77002-1014 63 Via Trieste, 35121 Padova USA ITALY [email protected] [email protected] xii Contributors

Ari Laptev Vol. I Mario Milman Vol. I Imperial College London Florida Atlantic University Huxley Building, 180 Queen’s Gate Boca Raton, Fl. 33431 London SW7 2AZ USA UK [email protected] [email protected] and Michele Miranda Jr. Vol. I Royal Institute of Technology University of Ferrara 100 44 Stockholm via Machiavelli 35 SWEDEN 44100, Ferrara [email protected] ITALY [email protected] Suzanne Lenhart Vol. III Dorina Mitrea Vol. III University of Tennessee Knoxville, TN 37996-1300 University of Missouri at Columbia USA Columbia, MO 65211 [email protected] USA [email protected] Vitali Liskevich Vol. II Marius Mitrea Vol. III Swansea University University of Missouri Singleton Park at Columbia Swansea SA2 8PP Columbia, MO 65211 UK USA [email protected] [email protected] Juan J. Manfredi Vol. II Stanislav Molchanov Vol. III University of Pittsburgh University of North Carolina Pittsburgh, PA 15260 at Charlotte USA Charlotte, NC 28223 [email protected] USA Moshe Marcus Vol. I [email protected] Israel Institute of Technology-Technion Sylvie Monniaux Vol. III 33000 Haifa Universit´eAix-Marseille 3 ISRAEL F-13397 Marseille C´edex20 [email protected] FRANCE Joaquim Mart´ın Vol. I [email protected] Universitat Aut`onomade Barcelona Vitaly Moroz Vol. II Bellaterra, 08193 Barcelona SPAIN Swansea University [email protected] Singleton Park Swansea SA2 8PP Eric Mbakop Vol. I UK Worcester Polytechnic Institute [email protected] 100 Institute Road Worcester, MA 01609 Umberto Mosco Vol. I USA Worcester Polytechnic Institute [email protected] 100 Institute Road Nicolas Meunier Vol. II Worcester, MA 01609 USA Universit´eParis Descartes (Paris V) [email protected] 45 Rue des Saints P`eres 75006 Paris Oleg Motygin Vol. II FRANCE Institute for Problems [email protected] in Mechanical Engineering Emanuel Milman Vol. I Russian Academy of Sciences Institute for Advanced Study V.O., Bol’shoy pr. 61 Einstein Drive, Simonyi Hall 199178 St. Petersburg Princeton, NJ 08540 RUSSIA USA [email protected] [email protected] Authors xiii

Nikolai Nadirashvili Vol. II Grigori Rozenblum Vol. III Centre de Math´ematiques Chalmers University of Technology et Informatique University of Gothenburg Universit´ede Provence S-412 96, Gothenburg 39 rue F. Joliot-Curie SWEDEN 13453 Marseille Cedex 13 [email protected] FRANCE [email protected] Yuri Safarov Vol. III King’s College London Yuri Netrusov Vol. III Strand, London WC2R 2LS University of Bristol UK University Walk [email protected] Bristol BS8 1TW UK Laurent Saloff-Coste Vol. I [email protected] Cornell University Mallot Hall, Ithaca, NY 14853 Serge Nicaise Vol. I USA Universit´eLille Nord de France [email protected] UVHC, 59313 Valenciennes Cedex 9 Evariste FRANCE Sanchez-Palencia Vol. II [email protected] Universit´ePierre et Marie Curie 4 place Jussieu Dian K. Palagachev Vol. III 75252 Paris Technical University of Bari FRANCE Via E. Orabona 4 [email protected] 70125 Bari ITALY Ren´eL. Schilling Vol. I [email protected] Technische Universit¨atDresden Institut f¨urStochastik Grigory P. Panasenko Vol. II D-01062 Dresden University Jean Monnet GERMANY 23, rue Dr Paul Michelon [email protected] 42023 Saint-Etienne FRANCE Gunther Schmidt Vol. II [email protected] Weierstrass Institute of Applied Analysis and Stochastics LuboˇsPick Vol. III Mohrenstr. 39, 10117 Berlin Charles University GERMANY Sokolovsk´a83, 186 75 Praha 8 [email protected] CZECH REPUBLIC [email protected]ff.cuni.cz Robert Seiringer Vol. I Princeton University Sergei V. Poborchi Vol. II P. O. Box 708 St. Petersburg State University Princeton, NJ 08544 28, Universitetskii pr., Petrodvorets USA St. Petersburg 198504 [email protected] RUSSIA [email protected] Christina Selby Vol. I Wolfgang Reichel Vol. II The Johns Hopkins University 11100 Johns Hopkins Road Universit¨atKarlsruhe (TH) Laurel, MD 20723. D-76128 Karlsruhe USA GERMANY [email protected] [email protected] J¨urgenRoßmann Vol. II Nageswari Shanmugalingam Vol. I University of Rostock Institute of Mathematics University of Cincinnati D-18051 Rostock Cincinnati, OH 45221-0025 GERMANY USA [email protected] [email protected] xiv Contributors

Johannes Sj¨ostrand Vol. III Alexander Ukhlov Vol. I Universit´ede Bourgogne Ben Gurion University of the Negev 9, Av. A. Savary, BP 47870 P.O.B. 653, Beer Sheva 84105 FR-21078 Dijon C´edex ISRAEL and UMR 5584, CNRS [email protected] FRANCE [email protected] Boris Vainberg Vol. III University of North Carolina Igor I. Skrypnik Vol. II at Charlotte Institute of Applied Charlotte, NC 28223 Mathematics and Mechanics USA Donetsk [email protected] UKRAINE [email protected] Igor E. Verbitsky Vol. III University of Missouri Ruxandra Stavre Vol. II Columbia, Missouri 65211 Institute of Mathematics USA “Simion Stoilow” Romanian Academy [email protected] P.O. Box 1-764 014700 Bucharest Gregory C. Verchota Vol. II ROMANIA Syracuse University [email protected] 215 Carnegie Nikos Stylianopoulos Vol. II Syracuse NY 13244 USA University of Cyprus [email protected] P.O. Box 20537 1678 Nicosia Laurent V´eron Vol. I CYPRUS Universit´ede Tours [email protected] Parc de Grandmont Susanna Terracini Vol. III 37200 Tours FRANCE Universit`adi Milano Bicocca [email protected] Via Cozzi, 53 20125 Milano ITALY Gr´egoryVial Vol. II [email protected] IRMAR, ENS Cachan Bretagne Achilles Tertikas Vol. I CNRS, UEB 35170 Bruz University of Crete and FRANCE Institute of Applied and [email protected] Computational Mathematics 71110 Heraklion Jie Xiao Vol. I GREECE Memorial University [email protected] of Newfoundland Jesper Tidblom Vol. I St. John’s, NL A1C 5S7 CANADA The Erwin Schr¨odingerInstitute [email protected] Boltzmanngasse 9 A-1090 Vienna Boguslaw Zegarlinski Vol. I AUSTRIA Imperial College London [email protected] Huxley Building S´ebastienTordeux Vol. II 180 Queen’s Gate London SW7 2AZ Institut de Math´ematiquesde Toulouse UK 31077 Toulouse [email protected] FRANCE [email protected] http://www.springer.com/978-1-4419-1342-5