0
INTERNATIONAL WORKSHOP
on
QUANTUM SPECTRA AND DYNAMICS
28 June – 5 July, 2000
Institute of Theoretical Physics
Technion – IIT
and
Einstein Institute of Mathematics
&
Edmund Landau Center for Research in Mathematical Analysis
& Related Areas
Hebrew University of Jerusalem
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THURSDAY 29TH JUNE
09:00-09:50
Claude-Alain Pillet: University of Toulon
Spectral theory of nonequilibrium steady states
09:50-10:30
Coffee
10:30-11:20
Barry Simon: Caltech
Approach to equilibrium in a forced burgers equation
11:30-12:20
Alexander Figotin: UC at Irvine
Nonlinear phenomena in periodic dielectric media
12:20-13:30
Lunch
13:30-14:20
Lorenzo Sadun: University of Texas, Austin
Geometry of adiabatic quantum pumps
14:30-15:20
Robert Schrader: FU Berlin
Kirchhof's rule for quantum wires
15:20-16:00
Coffee
16:00-16:50
Bernard Helffer: Universite Paris Sud
On semiclassical problems occurring in
superconductivity
17:00-17:50
Folkmar Bornemann: Munich University of
Technology
The method of weak convergence and adiabatic theorem
without a gap condition
PROGRAMME
Wednesday 28th June
09:30-10:20
10:20-10:30
10:30-11:20
11:30-12:20
12:20-13:30
13:30-14:20
14:30-15:20
15:20-16:00
16:00-16:50
17:00-17:50
Registration
Welcome
Jan Philip Solovej: University of Copenhagen
The correlation energy of charged quantum systems
Ofer Zeitouni: Technion
Quenched and annealed relations for random walks in
random environments
Lunch
Heinz Siedentop: University of Munich
On the renormalization of the energy of the electronpositron field
Uzy Smilansky: Weizmann Institute
Edge states in magnetic billiards: exterior-interior
Coffee
Israel Michael Sigal: University of Toronto
Quantum mechanics at positive temperatures
David Damanik: Caltech
Application of combinatorics on words to spectral
theory
21:00
19:00-21:00
Yosi Avron at Home
1
Conference Dinner in Atlit
2
FRIDAY 30TH JUNE
09:00-09:50
Alain Joye: University of Grenoble
Semiclassical dynamics and exponential asymptotics
09:50-10:30
Coffee
10:30-11:20
George Hagedorn: Virginia Polytechnic Institute and
State University
A review of rigorous results on molecular propagation
11:30-12:20
Gian Michele Graf: ETH
Extended edge states in quantum Hall systems
12:20-13:30
Lunch
13:30
SUNDAY 2ND JULY
09:00-09:50
Giovanni Gallavotti: University of
Rome, "La Sapienza"
tba
09:50-10:30
Coffee
10:30-11:20
Leonid Pastur: Universite Paris 7
Random matrices related to unitary group
11:30-12:20
Alexander Kamenev: Technion
Wigner-Dyson statistics in the replica field theory
12:20-13:30
Lunch
13:30-14:20
Peter Stollmann: University of Frankfurt
Multi-scale analysis implies strong dynamical
localization
14:30-15:20
Abel Klein: University of California at Irvine
A bootstrap multiscale analysis for random operators
and applications
15:20-16:00
Coffee
16:00-16:50
Peter Hislop: University of Kentucky
The Wegner estimate and the density of states
Free Afternoon
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3
MONDAY 3RD JULY
09:00-09:50
Yulia Karpeshina: University of Alabama at
Birmingham
Perturbation formulae for the two-dimensional periodic
magnetic Schrödinger operator
09:50-10:30
Coffee
10:30-11:20
Peter Kuchment: Wichita State University
Complex analysis in spectral problems for periodic
Hamiltonians
11:30-12:20
Shmuel Agmon: Hebrew University of Jerusalem
On eigenvalues and resonances of holomorphic operator
functions
12:20-13:30
Lunch
13:30-14:20
Ruedi Seiler: Technical University of Berlin
Bethe Ansatz solutions for Hotstadter type models
14:30-15:20
Francois Germinet: Universite de Lille 1 (USTL)
The spreading of wave-packets in quantum dynamics : a
nonlinear lower bound
15:20-16:00
Coffee
16:00-16:50
Serguie Tcheremchantsev: University of Orleans
The non-spreading of wave-packets in quantum
dynamics: dynamical localization
19:00-21:00
Conference Dinner at the Park Plaza Hotel
TUESDAY 4TH JULY
09:00-09:50
Gunter Stolz: University of Alabama at Birmingham
The Bernoulli-Anderson model
09:50-10:30
Coffee
10:30-11:20
Werner Kirsch:
Spectral theory for sparse potentials
11:30-12:20
Stanislav Molchanov: University of N. Carolina at
Charlotte
Schroedinger operators with random sparse potentials
12:20-13:30
Lunch
13:30-14:20
Svetlana Jitomirskaya: UC at Irvine
Non-perturbative ac spectrum for quasiperiodics
one-dimensional Schroedinger operators
14:30-15:20
Alexander Kiselev: University of Chicago
WKB asymptotics of generalized eigenfunctions for onedimensional Schrödinger operators
15:20-16:00
Coffee
16:00-16:50
Christian Remling: Ludwig Maximillians University
Schroedinger operators with decaying potentials
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4
WEDNESDAY 5TH JULY
09:00-09:50
Zeev Rudnik: Tel Aviv University
Quantum ergodicity for torus maps
09:50-10:30
Coffee
10:30-11:20
Frederic Klopp: Universite Paris-Nord
The monodromy matrix, the adiabatic wkb method and
the spectrum of quasi-periodic operators
on the real line.
11:30-12:20
Michael Aizenman: Princeton University
The statistical mechanics of Anderson localization
12:20-13:30
Lunch
13:30
End of Workshop
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THE METHOD OF WEAK CONVERGENCE AND ADIABATIC
THEOREMS WITHOUTA GAP CONDITION
ABSTRACTS
Folkmar Bornemann
Munich University of Technology
ON EIGENVALUES AND RESONANCES OF HOLOMORHIC
OPERATOR FUNCTIONS
We will present a new method of proof for adiabatic results in quantum theory
based on weak convergence in spaces of operator valued functions. This way the
adiabatic theorem without a gap condition and semi-classical limits of quantumclassical models in quantum chemistry can be treated in a unified way. We will
also adress the connection of our basic technical tool, a weak version of the
hypervirial theorem, to the notion of semi-classical or Wigner measure.
Shmuel Agmon
Hebrew University of Jerusalem
We shall discuss perturbation formulas for (generalized) eigenvalues and
resonances of holomorphic operator functions. Applications to embedded
eigenvalues will be described.
APPLICATION OF COMBINATORICS ON WORDS TO SPECTRAL
THEORY
THE STATISTICAL MECHANICS OF ANDERSON LOCALIZATION
David Damanik
California Institute of Technology
Michael Aizenmann
Princeton University
We discuss recent developments in the spectral theory of one-dimensional
quasicrystal models, that is, discrete one-dimensional Schrödinger operators with
hierarchical potentials of low combinatorial complexity. Primary examples are
given by Sturmian potentials and potentials generated by primitive substitutions.
Typical results are presented and the core ingredients to the proof strategies are
discussed. We explain, in particular, that the essential part of the analysis
consists of a detailed investigation of the combinatorial structure of a given
potential.
We shall discuss some recent advances(*) in the mathematical analysis of the
localization phenomena exhibited by operators with random potentials, as well as
some related conjectures, which draw on analogies with the analysis of phase
transitions in some basic models of statistical mechanics. [(*) Joint work with J.
Schenker, R. Friedrich, and D. Hundertmark.]
The analysis uses the fractional-moment decay condition as a convenient
signature of localization. The results enable to conclude, for a wide range of
discrete models: a strong form of dynamical localization, Poisson statistics for
the eigenvalues, and other aspects of localization, over the entire regime for
which localization follows by any of the currently available constructive criteria
(including the multiscale method). Also derived are some power-law lower
bounds on the decay of the resolvent at any mobility edge.
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NONLINEAR PHENOMENA IN PERIODIC DIELECTRIC MEDIA
EXTENDED EDGE STATES IN QUANTUM HALL SYSTEMS
Alexander Figotin
University of California at Irvine
Gian Michele Graf
ETH
A classical electron in a 2-dimensional domain  subject to a homogeneous
magnetic field moves in a circle about some fixed center --- the guiding center --as long as it does not hit the boundary . If it does, the electron follows the
boundary by repeatedly bouncing at it. The quantum mechanical counterpart
should be given by the existence of extended edge states. This idea, first put
forward by Halperin, is of some importance in connection with the quantum Hall
effect. We prove the existence of absolutely continuous edge spectrum at
energies away from the Landau levels, first for  a half--plane and then for more
general domains. The result is stable under addition of some disorder potential.
The proof is based on Mourre theory. The conjugate operator, i.e., the operator
increasing in time, is related to the position of the guiding center. This is joint
work with J. Fröhlich and J. Walcher.
We develop a consistent mathematical theory of weakly nonlinear periodic
dielectric media for the dimensions one, two and three. The theory is based on
the Maxwell equations with classical quadratic and cubic constitutive relations.
In particular, the theory gives a complete classification of different nonlinear
interactions between Floquet-Bloch modes based on indices which measure the
strength of the interactions. The indices take on only a relatively small number of
prescribed values collected in a table. The theory is based on the asymptotic
analysis of oscillatory integrals describing the nonlinear interactions.
THE SPREADING OF WAVE-PACKETS IN QUANTUM DYNAMICS : A
NONLINEAR LOWER BOUND
Francois Germinet
Universite de Lille 1 (USTL)
A REVIEW OF RIGOROUS RESULTS ON MOLECULAR
PROPAGATION
Understanding what determines the spreading of wave-packets arises as an issue
of the most interest in the last decade, motivated by numerical computations on
quasiperiodic models. Guarneri's ideas, pushed further by Combes and Last,
allowed one to get a lower bound on the dynamics expressed in terms of fractal
properties (like the Hausdorff measure) of the spectral measure associated to an
initial state . However numerical computations suggest that more subtle fractal
properties should be involved. In a joint work with J.-M. Barbaroux and S.
Tcheremchantsev, we derive a more refined lower bound that takes into account
George Hagedorn
Virginia Polytechnical Institute and State University
the generalized fractal dimensions
The time-dependent Born-Oppenheimer Approximation is the main source of
information about molecular propagation. It leads to an asymptotic expansion in
powers of  for solutions to the molecular Schrödinger equation. Here  is the
electron mass divided by the mean nuclear mass. By applying an optimal
truncation technique to this expansion, we obtain an approximation whose errors
4
D  (q) of the spectral measure   , and
are bounded by C1 exp( C 2 /  ) . A basic assumption of Born-Oppenheimer
Approximations is that the electron energy level of interest stay well separated
from the rest of the electronic spectrum. The simplest violations of this
assumption occur at crossings and avoided crossings. We describe the effects of
these phenomena on the propagation of molecular wave packets.
2
supply rigourous results concerning the basic properties of these generalized
fractal dimensions.
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7
ON SEMICLASSICAL PROBLEMS OCCURRING IN
SUPERCONDUCTIVITY
WIGNER-DYSON STATISTICS IN THE REPLICA
FIELD THEORY
Bernard Helffer
Universite Paris Sud
Alex Kamenev
Technion
In the analysis of minimizers for a Ginzburg-Landau functional, a Schrodinger
operator with magnetic field plays an important role. This leads in particular to
new questions about the localization of ground states in connection with the old
problem of the magnetic bottles.
Wigner - Dyson random matrix theory (RMT) appears to be the simplest (zerodimensional) case of the field theoretical treatment of a disordered media. Such
deterministic field theory of random phenomena may be constructed employing
either supersymmetry or replica tricks. The latter requires consideration of nonlinear action written in terms of n times n matrices, where in the end of the
calculation n-->0. We succeed calculating the corresponding functional integral
in terms of multiple saddle points expansion. The central objects in this
procedure are volumes of degenerate cosset space manifolds U(n)/U(p)U(n-p),
where p=0,1,...n. Subsequent analytical continuation n-->0 leads to the exact
RMT results. However the detailed mathematical nature of this analytical
continuation is not fully understood and requires more mathematical insight.
THE WEGNER ESTIMATE AND THE DENSITY OF STATES
Peter Hislop
University of Kentucky
A new proof of the Wegner estimate with the optimal volume dependence is
presented. The proof relies on new estimates for the spectral shift function. This
proof implies the existence of the density of states for some new random models.
This is joint work with J. M. Combes and S. Nakamura.
PERTURBATION FORMULAE FOR THE TWO-DIMENSIONAL
PERIODIC MAGNETIC SCHRÖDINGER OPERATOR
SEMICLASSICAL DYNAMICS AND EXPONENTIAL ASYMPTOTICS
Yulia Karpeshina
University of Alabama at Birmingham
Alain Joye
University of Grenoble
Asymptotic formulae for eigenvalues and eigenfunctions of the two-dimensional
periodic magnetic Schrödinger operator are constructed in the high-energy
region for a rich set of quasimomenta. The proof of the Bethe-Sommerfeld
conjecture is a simple corollary of these formulae.
We will describe some joint work with G.Hagedorn concerning constructions of
approximate solutions to the time-dependent Schroedinger equation in the
semiclassical limit by means of wave packets which are localized in phase space.
Our main result is an approximation which, under appropriate analyticity
assumptions on the potential, agrees with exact solutions up to exponentially
small errors. Further properties of this basic result, such as its long time limit or
its scattering version, will also be discussed.
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SPECTRAL THEORY FOR SPARSE POTENTIALS
THE MONODROMY MATRIX, THE ADIABATIC WKB METHOD AND
THE SPECTRUM OF QUASI-PERIODIC OPERATORS
ON THE REAL LINE.
Werner Kirsch
University of Ruhr
Frederic Klopp
Universite Paris-Nord
This talk is about joint work with Dirk Hundertmark and Krishna Maddaly. We
consider Schrödinger operators whose potential is "sparse" in the sense that it
decays in some averaged sense at infinity, but does not go to zero pointwise. The
main example is a random potential with decaying randomness. We prove the
existence of both point spectrum and absolutely continuous spectrum for a large
class of such potentials.
The talk will be an overview of recent results obtained in joint work with A.
Fedotov on the spectral theory of quasi-periodic Schrödinger operators on the
real line. We will present the main object of our work, the monodromy matrix,
explain how it relates to spectral theory. We will explain how to compute the
monodromy matrix in the adiabatic limit. At last, we will describe the spectral
results thus obtained.
WKB ASYMPTOTICS OF GENERALIZED EIGENFUNCTIONS
FORONE-DIMENSIONAL SCHRÖDINGER OPERATORS
COMPLEX ANALYSIS IN SPECTRAL PROBLEMS FOR PERIODIC
HAMILTONIANS
Alexander Kiselev
University of Chicago
Peter Kuchment
Wichita State University
We will present a new technique which allows to prove WKB-type asymptotics
of generalized eigenfunctions (and, as a consequence, presence of the absolutely
continuous spectrum) for a class of one-dimensional Schrödinger operators.
Another application leads to some results of optimal nature on the preservation
of absolutely continuous spectrum for one-dimensional Stark operators with
decaying or smooth potentials. This is a joint work with M. Christ
Spectral theory of periodic operators has attracted a lot of attention recently. It
has been realized since papers published in 70s by J. Avron, S. Novikov, B.
Simon, L. Thomas, and C. Wilcox that complex analysis plays an important role
in these studies. This becomes even more evident now. The talk will contain
examples of complex analytic tools and problems ranging from generalized
Cauchy-Riemann equations, to irreducibility of analytic varieties, to
uniformization of Riemannian surfaces that naturally arise in spectral problems
for periodic operators (among which one can mention absolute continuity of the
spectrum, absence of embedded eigenvalues, Liouville theorems, and others).
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1.
2.
3.
SCHROEDINGER OPERATORS WITH RANDOM
SPARSE POTENTIALS
SPECTRAL THEORY OF NONEQUILIBRIUM
STEADY STATES
S. Molchanov
(UNCC)
Claude-Alain Pillet
University of Toulon
The sparse and slowly decaying potentials can describe some features of the
Anderson model with a small coupling, first of all the coexistence of a.c. and p.p.
spectral components. The talk (based on the joint activity with V.Jaksic and
B.Vainberg) will present
Extension of Kato theorem to the class of the lattice Schroedinger
operators with a sparse or slowly decaying random potentials
Structure of the extended states for such operators
Statistics and the dynamical localization for the point component
The algebraic approach to the dynamics of infinite quantum systems near thermal
equilibrium naturally leads to the spectral theory of a self-adjoint operator that
we called Liouvillean. Recent progresses on this subject have shown the
usefulness of this concept. For systems driven out of thermal equilibrium by
mechanical or thermodynamical forces, this method generally fails because even
the existence of a steady state becomes problematic. I shall explain how to adapt
the Liouvillean concept to this situation. As an illustration, I will discuss the
existence of a steady state in a simple model: The N-level atom coupled to
several reservoirs.
SCHROEDINGER OPERATORS WITH
DECAYING POTENTIALS
RANDOM MATRICES RELATED TO UNITARY GROUP
Christian Remling
Ludwig Maximmillians University
Leonid Pastur
University Paris 7
This talk deals with the spectral properties of one-dimensional Schroedinger
operators with decaying potentials. I will try to give an overview, emphasizing
(some of) the more recent developments.
We present an overview of recent results on asymptotic behavior of eigenvalue
distribution of random matrices of large order n, considering those random
matrix ensembles whose probability law includes, as an independent factor, the
Haar measure on the unitary group U(n) We first discuss results on various forms
of their limiting density of states, related to certain variational problems and to
recent results in free probability. Then we outline results on the universality
property of the local eigenvalue statistics, both in the bulk of the spectrum and at
its generic edges. We mention also certain related facts of analysis and spectral
theory.
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10
KIRCHHOF'S RULE FOR QUANTUM WIRES
Robert Schrader
FU Berlin
QUANTUM MECHANICS AT POSITIVE TEMPERATURES
Israel Michael Sigal
University of Toronto
We consider the quantum theory of a particle moving freely on a system of onedimensional wires (finite and half-infinite), which are connected in an arbitrary
way at a finite set of vertices. We characterize all boundary conditions at the
vertices which lead to local Kirchhof rules (=conservation of the quantum
currents) at each vertex. The resulting on-shell scattering matrix, composed of
the transmission and reflection amplitudes at the half-infinite ends, is given in
closed form in terms of the boundary conditions, the lengths of the internal wires
and the energy. The quantum version of Kirchhof's rule is now given as the
unitarity of this on-shell S-matrix. Cutting the system into two leads to a
composition rule for the corresponding S-matrices.
In this talk I will review some recent results and open problems in nonequilibrium quantum statistical mechanics.
APPROACH TO EQUILIBRIUM IN A FORCED
BURGERS EQUATION
Barry Simon
California Institute of Technology
ON THE RENORMALIZATION OF THE ENERGY OF THE
ELECTRON-POSITRON FIELD
Heinz Siedentop
Universitaet Regensburg
This describes joint work with W. Kirsch. We reduce approach to equilibrium
for a forced Burgers equation to semigroup estimates for Schrodinger operators.
I'll discuss the proof of the necessary estimates for a periodic force and discuss a
much more general conjecture.
In a joint work with E.H. Lieb, we consider the relativistic electron-positron field
interacting with itself via the Coulomb potential defined with the physically
motivated, positive, density-density quartic interaction. The more usual normalordered Hamiltonian differs from the bare Hamiltonian by a quadratic term and,
by choosing the normal ordering in a suitable, self-consistent manner, the
quadratic term can be seen to be equivalent to a renormalization of the Dirac
operator. Formally, this amounts to a Bogolubov-Valatin transformation, but in
reality it is non-perturbative, for it leads to an inequivalent, fine-structure
dependent representation of the canonical anticommutation relations. This nonperturbative redefinition of the electron/positron states can be interpreted as a
mass, wave-function and charge renormalization, among other possibilities, but
the main point is that a non-perturbative definition of normal ordering might be a
useful starting point for developing a consistent quantum electrodynamics.
THE CORRELATION ENERGY OF CHARGED QUANTUM SYSTEMS
Jan Philip Solovej
University of Copenhagen
Density functional theories have gained an enormous importance in
computational chemistry and condensed matter physics. The credo of density
functional theory is that the energy of a gas of charged particles is expressible in
terms of the charge density only. Although this may be correct from a very
abstract point of view the much more interesting fact is that the energy can
indeed be well approximated by a fairly simple functional of the density. I will
discuss some of what is known rigorously. In particular, I will discuss the
simplest model, jellium, in which the density is nearly constant. I shall review
what is known rigorously both for fermionic and for bosonic jellium. The result
for bosonic jellium is recent joint work with E. Lieb.
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QUENCHED AND ANNEALED RELATIONS FOR RANDOM WALKS
IN RANDOM ENVIRONMENTS
MULTI-SCALE ANALYSIS IMPLIES STRONG
DYNAMICAL LOCALIZATION
Ofer Zeitouni
Technion
Suppose that the integers are assigned i.i.d. random variables
Peter Stollman
University of Frankfurt
{ x } (taking
values in the unit interval), which serve as an environment. This environment
defines a random walk { X n } (called a RWRE) which, when at x, moves one
We discuss and explain a recent joint result with D. Damanik concerning
dynamical localization for random models.
step to the right with probability
 x , and one step to the left with probability
THE BERNOULLI-ANDERSON MODEL
1   x . Solomon (1975) determined the almost-sure asymptotic speed v
Gunter Stolz
University of Alabama at Birmingham
(=rate of escape) of a RWRE. Subsequent work provided limit theorems for the
RWRE { X n } . We describe recent results on the asymptotics of the probability
of rare events of the form
The Bernoulli-Anderson model is the most "singular" variant of the Anderson
model, in the sense that the random coupling constants which define the potential
take only two values. Coincidentally, this also makes it the most realistic model
of alloys. Singular distributions cause various difficulties in determining spectral
properties. As a result, localization for the Bernoulli-Anderson model has only
been fully understood in the discrete, one-dimensional case, where two different
approaches by Carmona/Klein/Martinelli and Shubin/Vakilian/Wolff,
respectively, are known. In joint work with Damanik and Sims we show how
ideas from scattering theory can be combined with the Carmona/Klein/Martinelli
approach to prove localization (exponential and dynamical) for one-dimensional
continuous Bernoulli-Anderson models with arbitrary compactly supported
single-site potentials. The talk will also be used to point out some open
problems.
where
P( X n / n  A) or P ( X n / n  A) , in the situation
P( X n / n  A)  n 0 ). These are related to certain Lyapunov
exponents, and the relation between quenched and annealed behaviour of these
exponents will be described. Rather drastically different behaviour will then be
described for RWRE on the (random) Galton-Watson tree.
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PARTICIPANTS’ ADDRESSES
Shmuel Agmon
Hebrew University of Jerusalem
Jerusalem, Israel
[email protected]
Michael Aizenmann
Department of Mathematics
Princton University
Princeton
USA
[email protected]
Joachim Asch
Centre de Physique theorique
CNRS, Luminy Case 907
F-13288 Marseille Cedex 9
France
[email protected]
Yosi Avron
Department of Physics
Technion
Haifa 32000
Israel
[email protected]
Jean-Marie Barbaroux
Laboratorie de Mathematiques
UMR 6629 du CNRS
Universite de Nantes
F-44322 Nantes Cedex 03
[email protected]
Folkmar Bornemann
Center of Mathematical Sciences
Munich University of
Technology
D-80290 Muchich
German
[email protected]
Olivier Bourget
Institut Fourier
BP74
38402 St. Martin D'Heres
France
[email protected]
Gregory Braverman
Max Planck Institute
Heidelberg
[email protected]
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Almut Burchard
Department of Mathematics
PO Box 400137
Charlottesville, VA 22904-4137
USA
[email protected]
David Damanik
Department of Mathematics
CalTech
Pasadena CA91125
USA
[email protected]
Rafael Rene Del Rio Castillo
IRSAMC
Lab. Physique Quantique
UMR 5626
Universite Paul Sabatier
31062 Toulouse Cedex France
[email protected]
Alexander Figotin
Department of Mathematics
University of California at Irvine
Irvine, CA 92697, USA
[email protected]
Giovanni Gallavotti
Dipartimento di Fisica
University of Rome,
"La Sapienza"
P. Le Moro 2
00185 Rome, Italy
[email protected]
Francois Germinet
Universite de Lille 1 (USTL)
UFR de Mathematiques
59655 Villeneuve d'Ascq
France
[email protected]
Gian Michele Graf
Institute of Theoretical Physics
ETH Honggerberg, HPZ G 7
CH-8093 Zurich
Switzerland
[email protected]
George Hagedorn
Department of Mathatics
460 McBryde Hall
Virginia Polytechnic Institute and
State University
Blacksburg, Virginia 24061-0123
USA
[email protected]
13
Bernard Helffer
Batiment 425
Departement de Mathematiques
Universite Paris Sud
91404 Orsay
France
[email protected]
Peter Hislop
Mathematics Department
University of Kentucky
Lexington KY 40506-0027
USA
Dirk Hundertmark
Department of Mathematics
CalTech
Pasadena CA91125
USA
[email protected]
Victor Ivrii
Department of Mathetics
University of Toronto
100 St. George Str.
Toronto, Ontario M5S 3G3
Canada
Yulia Karpechina
Department of Mathematics
University of Alabama
at Birmingham
Birmingham, Al 35294-1170
[email protected]
Alexander Kiselev
Department of Mathematics
University of Chicago
5734 S. University Ave.
Chicago Il 60637, USA
[email protected]
Marcus Klein
Department of Mathematics
University of Potsdam
[email protected]
Abel Klein
Department of Mathematics
University of California at Irvine
Irvine CA 92697-3875
[email protected]
Frederic Klopp
Departement de Mathimatique
LAGA, Institut Galilie
Universiti Paris-Nord
F-93430 Villetaneuse
France
[email protected]
Peter Kuchment
Department of Mathematics
Wichita State University
Wichita, KS 67260-003
USA
[email protected]
Michael Landrigan
Department of Mathematics
University of California at Irvine
Irvine, CA 92697-3875
USA
[email protected]
Yoram Last
Department of Mathematics
Hebrew University of Jerusalem
Jerusalem
Israel
[email protected]
[email protected]
[email protected]
Svetlana Jitomirskaya
Department of Mathematics
University of California
Irvine, CA 92697
Werner Kirsch
Franz-Becker-Weg 12
D59494 Soest
Germany
[email protected]
Alain Joye
Universite de Grenoble 1
Institut Fourier
B.P. 74
38402 Saint-Martin d'Heres
France
[email protected]
Rowan Killip
Department of Mathematics
California Institute
of Technology
Pasadena CA 91125
USA
[email protected]
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Daniel Lenz
Fachbereich Mathematik
J.W. Goethe Universitaet
60054 Frankfurt/Main
Germany
[email protected]
Stanislav Molchanov
Mathematics Department
University of N. Carolina
at Charlotte
Charlotte, NC 28223-0001
USA
[email protected]
Hatem Najar
Departement de Mathimatique
LAGA, Institut Galilie
Universiti Paris-Nord
F-93430 Villetaneuse
France
[email protected]
Katherine Naimark
Weizmann Institute of Science
Rehovot 76100
Israel
@weizmann.weizmann.ac.il
Leonid Pastur
Centre de Mathematiques de
Jussieu
Universite Paris 7 Denis Diderot
Case Postale 7012
F-75251 Paris Cedex o5, France
[email protected]
Claude-Alain Pillet
PHYMAT
Universite de Toulon
B.P. 132
F-83597 La Guarde Cedex
France
Christian Remling
Mathematisches Institut
Ludwig Maximillians University
Theresienstr. 39
80333 Munich,
Germany
[email protected].
unimuenchen.de
Thomas Richter
Technische Universietaet Berlin
Fachbereich Mathematik
Sekr, MA7-2
Strase des 17 Juni 136
D-10623 Berlin, Germany
[email protected]
Lorenzo Sadun
Department of Mathematics
University of Texas at Austin,
Austin TX 78712
USA
[email protected]
Robert Schrader
Institut fuer Theoretische Physik
Freie Universitaet Berlin
Arnimallee 14
14195 Berlin, Germany
[email protected]
Zeev Rudnick
Department of Mathematics
Tel Aviv University
Ramat Aviv
Israel
[email protected]
Ruedi Seiler
Technische Universitaet Berlin
Fachbereich Mathematik
Strasse des 17. Juni 136
D-10623 Berlin
Germany
[email protected]
Hermann Schulz-Baldes
University of California
Department of Mathematics
Irvine CA 92697-3875
USA
[email protected]
Heinz Siedentop
Mathematik
Universitaet Muenchen
Theresienstr. 39
80333 Muenchen
Germany
[email protected]
Israel Michael Sigal
Department of Mathematics
University of Toronto
100 St. George Str.
Toronto, Ontario M5S 1A1
Canada
[email protected]
Barry Simon
Mathematics Department
CalTech
Pasadena CA 91125
USA
[email protected]
[email protected]
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Uzy Smilansky
Physics Department
Weizmann Institute of Science
Rehovot 76100
Israel
[email protected]
Peter Stollmann
Department of Mathematics
University of Frankfurt
Robert-Mayer-Strasse 10
D-60325 Frankfurt am Main
Germany
Jan Philip Solovej
Department of Mathematics
University of Copenhagen
Universitetsparken 5
DK-2100 Copenhagen
Denmark
[email protected]
Gunter Stolz
UAB Department of Physics
CH452
Birmingham AL35294-1170
USA
[email protected]
[email protected]
Serguei Tcheremchantsev
University of Orleans
8 rue de la Gervaise
45470 Trainou
France
Serguei.Tcherem@labomath.
univ-orleans.fr
Ofer Zeitouni
Department of Mathematics
Technion
Haifa 32000
Israel
[email protected]
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