Partial Differential Operators and Mathematical Physics: International Conference in Holzhau, Germany, July 3’9, 1994

421 Pages·1995·10.495 MB·English

by  Sergio Albeverio, Fan Ru-Zong (auth.), Prof. Michael Demuth, Prof. Dr. Bert-Wolfgang Schulze (eds.)

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)
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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)
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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).