ICMP 2003

Quantum mechanics and spectral theory

Session organized by C. Gerard (Orsay), R. Weder (Mexico).


T. ChenNew York
P. ExnerPrague
T. IchinoseKanazawa
V. KostrykinAachen
E. SkibstedAarhus
G. StolzBirmingham, USA
D. YafaevRennes
V.A. ZagrebnovMarseille
Thomas Chen
Courant Institute
New York University

Localization lenghts and Boltzmann limit for the Anderson model at small disorder in d=3

We prove lower bounds on the localization lengths of eigenvectors in the 3-dimensional Anderson model, using an extension of techniques developed by L. Erdos and H.-T. Yau. This result is closely related to recent work of C. Shubin, W. Schlag and T. Wolff for dimensions 1 and 2. Furthermore, we demonstrate that the macroscopic limit of the corresponding lattice random Schrodinger system is governed by the linear Boltzmann equations.

Pavel Exner
Department of Theoretical Physics
NPI, Academy of Sciences
CZ-25068 Rez -- Prague
Czech Republic

Schrödinger operators with graph-type interactions

We consider a class of Schrödinger operators given formally by the expression (-i∇-A)2 - αδ(x-Γ), where Γ is a generalized graph, i.e. a subset of Rd, d=2,3, which can be decomposed into a union of curves and surfaces. We show how the geometry of Γ is reflected in the spectral properties and discuss related physical effects. Some open problems will also be listed.

T. Ichinose
Department of Mathematics
Kanazawa University

Quantum formula for quantum Zeno dynamics

The fact the decay can be slowed or even prevented by frequently repeated measurements, noticed first by Beskow and Nilsson, and called quantum Zeno effect by Misra and Sudarshan, attracted recently a lot of attention since it seems to be nowadays experimentally accessible. Mathematically the issue was addressed, in particular, long time ago by Friedman and Chernoff, however, the question about existence of the "Zeno limit" remained open.

Let H and P be a non-negative self-adjoint operator acting in a separable Hilbert space H and an orthogonal projection on H, respectively. Suppose that the operator HP := (H1/2P)*(H1/2P) is densely defined, then we claim that the said limit exists and we find its value,

limn→∞ [P exp(-iεtH/n)P]n = exp(-iεtHP) P

for ε=±1. In fact, the proof given in [1] yields a stronger result with P on the left-hand side replaced by values of a projection-valued family such that P(t)→P as t→0 and P(t)P = P(t), as well as non-symmetric versions of such formulae. The method we use employs a modification of a Kato's result on Trotter product formula in combination with analyic continuation.

As an example consider H=-Δ in L2(Rd) and P projecting onto an open Ω⊂Rd with a smooth boundary. Then the limits exists and HP is equal to the Dirichlet Laplacian in L2(Ω); this provided a rigorous proof of the formal claim made recently in [2] using the method of stationary phase.

[Joint work with Pavel Exner (Czech Academy of Sciences)]

  1. P. Exner, T. Ichinose, math-ph/0302060.
  2. P. Facchi et al., Phys. Rev. A 65 (2002), 012108.
Vadim Kostrykin
Fraunhofer-Institut für Lasertechnik
Steinbachstraße 15
Aachen, Germany

On Adiabatic Theorem of Quantum Mechanics

We study the variation of spectral subspaces under the Schrödinger evolution provided that the perturbation depends "slowly" on time. As a result we obtain a new proof of the Adiabatic Theorem of Quantum Mechanics and estimate the rate at wich the adiabatic limit is approached.

This is a joint work with K. A. Makarov (Missouri-Columbia).

E. Skibsted
MaPhySto and Institute of Mathematical Sciences
University of Aarhus
Ny Munkegade
8000 Aarhus C

Zero energy asymptotics of the resolvent in the long range case

We prove absence of eigenvalue at zero for two-body Schrödinger operators with long-range potentials having a positive virial at infinity. This enables us to prove a limiting absorption principle at zero and in fact a complete asymptotic expansion of the resolvent in weighted spaces in cones; one of the two branches of boundary for the cones is given by the positive real axis. The principal tools are singular Mourre theory and microlocal estimates.

[Joint work with S. Fournais (Laboratoire de Mathématiques Université Paris-Sud)]

Gunter Stolz
UAB, Department of Mathematics, CH452
AL 35294-1170

Fractional moment methods for Anderson localization in the continuum

The fractional moment (or Aizenman-Molchanov) method has been developed in the 90s as an alternative to multiscale analysis in the proof of Anderson localization. While yielding stronger results, the method was initially only applicable to lattice models.

We will present recent joint work with M. Aizenman, A. Elgart, S. Naboko and J. Schenker which allows to extend the fractional moment method and its consequences to continuum Anderson-type models. In the present talk we will focus on the new ideas from spectral theory and harmonic/complex analysis which enter the proofs. Other aspects of this work will be discussed in Elgart's talk in the session on Condensed Matter Physics.

D. Yafaev
Department of Mathematics
University of Rennes
Campus Beaulieu
35042 Rennes

A particle in a magnetic field of an infinite rectilinear current

We consider the Schrödinger operator H = (i∇+A)2 in the space L2(R3) with a magnetic potential A created by an infinite rectilinear current and perform its spectral analysis almost explicitly. In particular, we show that the operator H is absolutely continuous, its spectrum has infinite multiplicity and coincides with the positive half-axis. Then we find the large-time behavior of solutions exp(-iHt)f of the time dependent Schrödinger equation. Our main observation is that a quantum particle has always a preferable (depending on its charge) direction of propagation along the current. Similar result is true in classical mechanics.

Valentin A. Zagrebnov
Université de la Mediterranée (Aix-Marseille II)
and Centre de Physique Théorique-CNRS-Luminy-Case
907 13288 Marseille Cedex 9

Trotter-Kato product formula: recent results

Recently T. Ichinose, Hideo Tamura, Hiroshi Tamura, and V.A. Zagrebnov have proved the following result:

THEOREM: Let A and B be non-negative self-adjoint operators such that the operator sum C := A + B is self-adjoint in domain dom(C) = dom(A) ∩ dom(B).  Then:
(a) the Trotter product formulae converge in operator norm with error bound O(n-1):
||(e-tA/n e-tB/n)n - e-tC|| = O(n-1),     n→∞,
||(e-tB/2n e-tA/n e-tB/2n)n - e-tC|| = O(n-1),     n→∞,
uniformly on each compact t-interval in [0, ∞). Further, if C is strictly positive, then it holds uniformly on [0, ∞);
(b) the error bound O(n-1) is ultimate optimal.

These conditions on the generators have been formulated for the first time be E. Nelson. The hypothesis on the optimal rate of convergence discussed since 1993.

A recent result by V.A. Zagrebnov says that the same optimal estimate is valid in the trace-norm for Gibbs semigroups.


G.A. Al-Kader
Al-Azhar U., Cairo

Interaction of the superpositions of squeezed displaced Fock states with atomic systems

N. Artamonov
Moscow State U.

Localization of spectrum for certain classes of operator pencils

F. Brau
U. Mons-Hainaut, Belgium

Upper and lower limits for the number of bound states in an attractive central potential

N. Catarino, R. MacKay

Spectrum of quasiperiodic discrete Schrödinger operator

T. Chen
Courant Inst.

Localization of length and Boltzamn limit of the Anderson model at small disorders in d=3

D. Chruscinski
Nicholaus Copernicus

Spectral theory and quantum damped systems

N. Coftas

Hypergeometric type functions in a supersymmetric approach

R. Floreanini

Complete positivity from factorized dynamics

G. Gentile
Roma III

Quasi-periodic solutions and pure point spectrum for two-level systems

D. Hasler, F. Bernasconi, G. Graf
Copenhagen; ETH Zurich; ETH Zurich

The heat kernel expansion for the elctromagnetic field in a cavity and its application to the Casimir energy

P. Exner

Product formula for quantum Zeno dynamics

A. Iantchenko

On the positivity of the Jansen-Hess operators for arbitrary mass

T. Ichinose

Quantum formula for quantum zero dynamics

O. Kirillov
Moscow Lomonosov

How do small velocity dependent forces (de)stabilize a non-conservative system?

D. Krejcirik
IST, Lisbon

The nature of the essential spectrum in curved quantum waveguides

B. Kuckert

Large spin and statistics in non-relativistic quantum mechanics

J. Dittrich, J. Kirzy
NPI Reiz

Quantum waveguides with combined boundary conditions

R. Madrid

The role of the rigged Hilbert space in quantum mechanics

D. Marchetti
São Paulo

Spectral transitions in a class of off-diagonal Jacobi matrices: a probabilistic approach

H. Makino
Tokai, Kanagawa

Energy level statistics of integrable quantum systems based on the Mehta-Berry-Robink approach

M. Mekhfi
Lab. Phys. Math., Algeria

Topological quantum mechanics and Bessel functions

K. Nemcova
Czech Academy Sciences

Quantum mechanics of Dirichlet layers with a finite number of point perturbations

R. Parmar
Sandar Patel U., India

Study of partition functions using supersymmetric quantum mechanics

I. Popov
Leningrad Institute, St. Petersburg

Waveguides and layers coupled through small apertures: asymptotics of bound states, bands and resonances

O. Post, F. Lledo
IPAM, Aachen

Spectral gaps on manifolds with noncommutative group actions

N. Bogoliubov, U. Taner, A. Prykarpatsky
Steklov, Moscow; EMU, Famagusta; UMM Poland

On modeling of the Maxwell-Bloch quantum optical super-radiance proccess and its application to quantum computing

V. Kravchenko, M. Ramirez
IPN, Mexico

On a purely real quaternionic reformulation of Dirac equation

N. Roehrl

A new numerical method to solve the inverse Sturm-Liouville problem

M. Ramirez
IPN, Mexico

On a purely real quaternionic formulation of the Dirac equation

M. Rouleux
CPT, Marseille

Tunneling effects between tori in double wells

R. Schubert

Semiclassical time evolution for large times on manifolds of negative curvature

N. Stoilova

Wigner quantum systems: non-commutative sl(1|3) oscillator

A. Suzko
LIT, Dubna

Exactly solvable time-dependent Schrödinger equations

N. Ueki

The integrated density of states of random Pauli Hamiltonians

D. Yafaev

A particle in a magnetic field of an infinite rectilinear current

B. Zahkariev
LTP, Dubna

Applications of Fermi's golden rule to open quantum systems

V. Zagrebnov
CPT, Marseille

Trotter-Kato product formula: recent results

G. Garcia-Calderón
UNAM, Mexico City

Solution of the time-dependent Schrodinger equation for decay in terms of resonant states