Operator Theory Advances and Applications Vol. 78 Editor I. Gohberg Editorial Office: T. Kailath (Stanford) School of Mathematical H.G. Kaper (Argonne) Sciences S.T. Kuroda (Tokyo) Tel Aviv University P. Lancaster (Calgary) Ramat Aviv, Israel L.E. Lerer (Haifa) E. Meister (Darmstadt) Editorial Board: B. Mityagin (Columbus) J. Arazy (Haifa) V.V. Peller (Manhattan, Kansas) A. Atzmon (Tel Aviv) J.D. Pincus (Stony Brook) J.A. Ball (Blackburg) M. Rosenblum (Charlottesville) A. Ben-Artzi (Tel Aviv) J. Rovnyak (Charlottesville) H. Bercovici (Bloomington) D.E. Sarason (Berkeley) A. Bottcher (Chemnitz) H. Upmeier (Marburg) L. de Branges (West Lafayette) S.M. Verduyn-Lunel (Amsterdam) K. Clancey (Athens, USA) D. Voiculescu (Berkeley) L.A. Coburn (Buffalo) H. Widom (Santa Cruz) K.R. Davidson (Waterloo, Ontario) D. Xia (Nashville) R.G. Douglas (Stony Brook) D. Yafaev (Rennes) H. Dym (Rehovot) A. Dynin (Columbus) P.A. Fillmore (Halifax) Honorary and Advisory C. Foias (Bloomington) Editorial Board: P.A. Fuhrmann (Beer Sheva) P.R. Halmos (Santa Clara) S. Goldberg (College Park) T. Kato (Berkeley) B. Gramsch (Mainz) P.O. Lax (New York) G. Heinig (Chemnitz) M.S. Livsic (Beer Sheva) J.A. Helton (La Jolla) R. Phillips (Stanford) M.A. Kaashoek (Amsterdam) B. Sz.-Nagy (Szeged) Partial Differential Operators and Mathematical Physics International Conference in Holzhau, Germany July 3-9, 1994 Edited by M. Demuth B.-W. Schulze Birkhauser Verlag Basel· Boston· Berlin Volume Editorial Office: Prof. Michael Demuth Prof. Dr. Bert-Wolfgang Schulze Technische Universitat Clausthal Max Planck-Arbeitsgruppe Institut fiir Mathematik "Partielle Differentialgleichungen und Komplexe Analysis" Erzstrasse I Universitat Potsdam 38678 Clausthal-Zellerfeld Institut fiir Mathematik Germany Postfach 60 1553 14415 Potsdam Germany A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Deutsche Bibliothek Cataloging-in-Publication Data Partial differential operators and mathematical physics : international conference in Holzhau (Germany), July 3-9, 1994 / ed. by M. Demuth; B.-W. Schulze. -Basel; Boston; Berlin: Birkhiiuser, 1995 (Operator theory; Vol. 78) NE: Demuth, Michael [Hrsg.); GT 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 the permission of the copyright holder must be obtained. © 1995 Birkhauser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Softcover reprint of the hardcover 15 t edition 1995 Printed on acid-free paper produced from chlorine-free pulp Cover design: Heinz Hiltbrunner, Basel ISBN-13: 978-3-0348-9903-1 e-ISBN-13: 978-3-0348-9092-2 001: 10 .1 007/978-3-0348-9092-2 987654321 Contents Preface ..................................................................... ix S. Albeverio, F. Ru-Zong, M. Rockner, W. Stannat: A remark on coercive forms and associated semigroups ................ 1 W. Arendt, S. Monniaux: Domain perturbation for the first eigenvalue of the Dirichlet Schrodinger operator .................................................. 9 E. Bernardi, A. Bove: Geometric transition for a class of hyperbolic operators with double characteristics ................................................ 21 P. Boggiatto, E. Buzano, L. Rodino: Multi-quasi-elliptic operators in]Rn ................................... 31 L. Boutet de Monvel: Real analogue of the Bergman kernel ................................. 43 V.V. Chepyzhov, M.I.Vishik: Attractors of non-autonomous evolution equations with translation-compact symbols ......................................... 49 J. M. Combes, P. D. Hislop: Localization for 2-dimensional random Schrodinger operators with magnetic fields ........................................ 61 Yu. L. Daletskii, V. R. Steblovskaya: Some problems of calculus of variations in infinite dimensions ......... 77 M. Demuth, F. Gesztesy, J. van Casteren, Z. Zhao: Finite capacities in spectral theory ................................... 89 J. Derezinski: Classical N -body scattering .......................................... 99 P. Duclos, P. Stovicek: Quantum Fermi accelerators with pure-point quasi-spectrum ........ 109 Yu.V. Egorov, V.A. Kondratiev: On moments of negative eigenvalues of an elliptic operator .......... 119 L. Erdos: Magnetic Lieb-Thirring inequalities and stochastic oscillatory integrals ................................................. 127 B. V. Fedosov: On the trace density in deformation quantization .................... 133 D. Fujiwara: The stationary phase method with remainder estimate as dimension of the space goes to infinity . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135 VI Contents P. B. Gilkey, B. Botvinnik: The eta invariant, equivariant spin bordism, and metrics of positive scalar curvature ............................................ 141 J. Ginibre, G. Velo: Generalized Strichartz inequalities for the wave equation ............ 153 B. Helffer: Around the transfer operator and the 'Irotter-Kato formula .......... 161 R. Hempel, 1. Herbst: Bands and gaps for periodic magnetic Hamiltonians ................. 175 M. Hieber: If'" -calculus for second order elliptic operators in divergence form .................................................. 185 T. Ichinose: Path integral for the relativistic Schrodinger semigroup 191 V. Ivrii: Semiclassical spectral asymptotics and multiparticle quantum theory .................................................... 199 S. T. Kuroda, K. Kurata: Product formulas and error estimates ............................... 213 A. Laptev: On inequalities for the bound states of Schrodinger operators 221 M. Levitin, D. Vassiliev: Some examples of two-term spectral asymptotics for sets with fractal boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 227 O. Liess: Estimates for Fourier transforms of surface-carried densities on surfaces with singular points ..................................... 235 A. Martinez, G. Nenciu: On adiabatic reduction theory ...................................... 243 1. McGillivray: Recurrence for fractional powers of diffusion operators in terms of volume-growth .......................................... 253 S. Nakamura: Band spectrum for Schrodinger operators with strong periodic magnetic fields ............................................. 261 V. S. Rabinovich: Mellin pseudodifferential operators with operator symbols and its applications ................................................. 271 M. Reissig, K. Yagdjian: Hypoellipticity of certain differential operators with degeneration of infinite order ........................................ 281 Contents vii 1. Roitberg: On approximation of solutions of elliptic boundary value problems for Petrovskii elliptic systems by linear combinations of fundamental solutions .............................. 285 Y. Saito: The reduced wave operator with two unbounded media .............. 291 E. Schrohe, B.-W. Schulze: Mellin quantization in the cone calculus for Boutet de Monvel's algebra ......................................... 299 B.-W. Schulze: Transmission algebras on singular spaces with components of different dimensions .............................................. 321 Z. G. Sheftel: On approximation by solutions of non-local elliptic problems ........ 343 A. Shlapunov, N. Tarkhanov: A stability set in the Cauchy problem for elliptic systems ............ 353 A. V. Sobolev: Discrete spectrum asymptotics for the Schrodinger operator 357 P. Stollmann: Convergence of Schrodinger operators on varying domains 369 G. Stolz: Localization for the Poisson model .................................. 375 c. A. Tracy, H. Widom: Systems of partial differential equations for a class of operator determinants ............................................ 381 J. Voigt: Absorption semigroups, Feller property, and Kato class .............. 389 L. Weis: Gaussian estimates and analytic semigroups ......................... 397 D. Yafaev: New channels of scattering for long-range potentials ................. 405 Participants ............................................................... 421 Talks ..................................................................... 427 Preface This volume contains the proceedings of the International Conference on "Par tial Differential Equations" held in HolzhaujErzgebirge, Germany, July 3~9, 1994. The conference was sponsored by the Max-Planck-Gesellschaft, the Deutsche For schungsgemeinschaft, the Land Brandenburg and the Freistaat Sachsen. It was initiated by the Max-Planck-Research Group "Partielle Differential gleichungen und Komplexe Analysis" at the University of Potsdam as one of the annual meetings of the research group. This conference is part of a series begun by the former Karl-Weierstraf3-Institute of Mathematics in Berlin, with the confer ences in Ludwigsfelde 1976, Reinhardsbrunn 1985, Holzhau 1988 (proceedings in the Teubner Texte zur Mathematik 112, Teubner-Verlag 1989), Breitenbrunn 1990 (proceedings in the Teubner Texte zur Mathematik 131, Teubner-Verlag 1992), and Lambrecht 1991 (proceedings in Operator Theory: Advances and Applications, Vol. 57, Birkhiiuser Verlag 1992); subsequent conferences took place in Potsdam in 1992 and 1993 under the auspices of the Max-Planck-Research Group "Partielle Differentialgleichungen und Komplexe Analysis" at the University of Potsdam. It was the intention of the organizers to bring together specialists from differ ent areas of modern analysis, geometry and mathematical physics to discuss not only recent progress in the respective disciplines but also to encourage interaction between these fields. The scientific advisory board of the Holzhau conference consisted of S. Al beverio (Bochum), L. Boutet de Monvel (Paris), M. Demuth (Clausthal), P. Gilkey (Eugene), B. Gramsch (Mainz), B. Helffer (Paris), S.T. Kuroda (Tokyo), B.-W. Schulze (Potsdam). The organizers would like to thank Frau Chr. Gottschalkson for executing a great part of the administrative work as well as Frau Albrecht from the Max Planck-Institute of Mathematics in Bonn. M. Demuth B.-W. Schulze Clausthal Potsdam Operator Theory: Advances and Applications, Vol. 78 © 1995 Birkhauser Verlag Basel/Switzerland A Remark on Coercive Forms and Associated Semigroups S. Albeverio, F. Ru-Zong, M. R6ckner, w. Stannat Abstract It is shown how to determine a coercive closed form on a real Hilbert space directly from its associated semigroup. AMS Subject Classification (1991) Primary: 31C25 Secondary: 47D06 1 Introduction The abstract theory of (real valued) coercive closed forms on real Hilbert spaces and associated semigroups is presented in [1], with the aim of application in the theory of Dirichlet forms. In particular, the relation between such forms, semi groups, associated resolvents and generators is studied. However, in contrast to the symmetric case, a direct connection between semigroups and coercive closed forms is missing. The purpose of this paper is to deliver this relation, complement ing Diagram 3 in [1] this way. In particular, we show how one can construct the coercive closed form (including its domain) directly from the associated semigroup. In fact, one part of this result (i.e., Theorem 3.4(ii) below) in the case of complex valued coercive closed forms can already be found in [3]. But for completeness we also give a proof of this in the real case, in particular adding the details of the un derlying results from [5] which are not spelt out in the latter reference. We use the characterization of (£,D(£)) in terms of its associated resolvent (cf. Proposition 2.5 below) in an essential way. We also discuss some consequences. Let us indicate shortly our results. Let (£, Vee)) be a coercive closed form on a real Hilbert space 1t with associated semigroup (Tdt>o. For t E (0,00) let (t)£(u,v):= t(u-Ttu,v); u,v E 1t. We shall prove in Theorem 3.4: u E Vee) {::::::} sup (t)£(u,u) < 00 t>O and £(u,v) = lim (t)£(u,v) for all u,v E Vee). ttO 2 S. Albeverio, F. Ru-Zong, M. Rockner, W. Stannat Moreover, the restriction of (Tt}t>o to V(£) is a strongly continuous semigroup on V(£). In particular lim£l(u - Ttu,u - Ttu) = 0 for all u E V(£). tlO In Theorem 4.1 we shall give another characterization of V( £) by using a Tauberian theorem. For u E 1i define au(t):= J~(u - Tsu,u)ds; t E (0,00). Then u E V(£) <===} 3 lim ~au(t) E R tlO t and lim ~au(t) = !£(u,u) for all u E V(£). tlO t 2 2 Definitions and Preliminaries Let 1i be a real Hilbert space with inner product (.,.) and norm 11·11 = (.,.) ~. Let V(£) be a linear subspace of 1i and £ : V(£) x V(£) ---+ R a non-negative definite bilinear map. For a ~ 0 we set £a(u, v) := £(u, v) + a(u,v); u,v E V(£). (2.1) Definition 2.1. (£, V(£» is said to satisfy the weak sector condition iff there exists a constant K > 0 such that (2.2) The constant K in Definition 2.1 is called continuity constant. It is easy to see that the weak sector condition is equivalent to the following: there exists a constant K' > 0 such that (2.3) Definition 2.2. (£, V(£» is said to be closed if V(£) is complete with respect to the norm £-11 /2 ,where £-(u,v) := 1/2 ( £(u,v) + £(v,u) ) ; u,v E V(£). Definition 2.3. (£, V(£» is said to be a coercive closed form iff V(£) is a dense subspace of 1i, and (£, V(£) is closed and satisfies the weak sector condition (2.2).