Download Quantum Few-Body Systems

Quantum Few-Body Systems
University of Aarhus
March 19–20, 2007
Speakers and Abstracts
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Jean-Michel Combes, Université de Toulon et du Var, France
“Recent advances and open problems in the Mathematical Theory of
Random Schrödinger Operators”
We review the present status of the Quantum One Body Theory of
localization and transport in the Anderson model. In particular we present
recent results on the regularity properties of the Density of States which
play a basic role through the so-called "Wegner estimates" in the multiscale analysis à la Fröhlich-Spencer.
Horia Cornean, Aalborg University, Denmark
“An elementary derivation of the Brueckner-Goldstone groundstate energy
expansion”
J. Goldstone used the Gell-Mann & Low Theorem and Feynman diagrams
in order to obtain the so called linked-cluster perturbation formula (see J.
Goldstone: "Derivation of the Brueckner Many-Body Theory",
Proceedings of the Royal Society of London. Series A, Mathematical and
Physical Sciences, Vol. 239, No. 1217. (Feb. 26, 1957), pp. 267-279.). We
reobtain this result by using time-independent analytic perturbation theory.
Jozef T. Devreese, Universiteit Antwerpen, Belgium
“On the Response Properties of Fröhlich Polarons”
Fröhlich polarons constitute a field-theoretical model with a richness of
realisations in condensed matter systems. For example, optical and
magneto-optical properties of polar media, including systems of reduced
dimension and reduced dimensionality, are influenced by polaron effects.
Fröhlich polarons have been investigated using i. a. perturbation schemes,
canonical transformations, adiabatic approach and Feynman path integrals.
Of particular interest are few- and many- polaron systems. Whereas the
single Fröhlich-polaron problem has been solved to a large degree (and the
recent diagrammatic quantum Monte Carlo studies provide a convenient
test), the few- and many polaron problems retain secrets. The Fröhlichpolaron concept has also been extended or modified e. g. to allow for the
study of small (or Holstein-) polarons, piezopolarons, electronic polarons,
bipolarons, polaronic excitons… In this contribution I analyse our present
understanding of the polaron problem and examine possible future
prospects.
References:
J.T. Devreese, “Polarons and Bipolarons in Nanostructures” in Handbook
of Semiconductor Nanostructures and Nanodevices. P. 311. (Edited by
A.A. Balandin and K.L. Wang, ASP, Cal. USA, 2006).
Ibid: cond-mat/0607121, “Fröhlich Polarons from 3D to 0D. Concepts and
Recent Developments”.
Speaker
Nils Elander, Ksenia Shilyaeva and Mikhail Volkov, AlbaNova University
Center, Dept. of Physics, Stockholm University
Title
“The Resonances wave function: is it relevant? - Applications in Molecular
Physics”
A resonance is often associated with a structure in a cross section through
the Breit-Wigner formalism. Here we try to describe how, the in principle
unphysical, resonance wave function can be used to describe physical
phenomena. Using the complex dilation formalism we discuss how the
resonance state appears in the description of fragmentation and scattering
processes.
Abstract
After a breif description of our numerical methods we present a few
applications two and three-atomic systems are presented.
Speaker
Pavel Exner, Doppler Institute for Mathematical Physics and Applied
Mathematics, Prague
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“Quantum networks modelled by graphs”
Quantum networks are often modelled using SchrÄodinger operators on
metric graphs. To give meaning to such models one has to know how to
interpret the boundary conditions which match the wave functions at the
graph vertices. In this talk we discuss several approximation results which
serve this purpose. One group of them concerns approximation by means
of "fat graphs", in other words, suitable families of shrinking manifolds; we
discuss convergence of the spectra and resonances in such a setting. The
other possibility is to use families of interactions on the graph itself, or
more generally, families of graphs with vertices and edges added. We will
show a way how one can approximate in this way a generic time-reversal
invariant vertex coupling.
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Ludwig D. Faddeev, Steklov Institute of Mathematics, Sankt Petersburg
“The derivation of Efimov Effect via integral equations.”
Recently we witness return of interest to Efimov Effect due to new
experiments. In my talk I describe the resons, based on the three body
scattering integral equations, which I produced to support Efimov' claim,
when he came to consult with me 35 years ago.
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Jérémy Faupin, Université de Reims, France
“Existence of a ground state for the confined hydrogen atom in nonrelativistic QED”
We consider a system of one nucleus and one electron interacting with the
quantized electromagnetic field. Instead of fixing the nucleus, we assume
that the center of mass of the system is confined by a potential. We prove
that some binding conditions are satisfied, which implies the existence of a
ground state for the Hamiltonian associated with the model.
Dmitry K. Gridnev, Kassel University, Germany
“Critically Stable Systems”
Recent developments in the analysis of critically stable systems are
discussed. The behavior of weakly bound states in many-particle systems
as they approach the continuum threshold is analyzed in the framework of
non--relativistic quantum mechanics. Under minor assumptions it is proved
[1] that if the lowest dissociation threshold is the decay channel into
likewise (all positively or all negatively) charged clusters then the bound
state, which approaches the threshold, does not spread and eventually
becomes the bound state at the threshold. In such systems this means
effectively super-size blocking, i.e. loosely bound particles do not form
extended halo-like structures. The results have a direct application to halo
nuclei and negative atomic and molecular ions. A positive proof is given to
the conjecture that negative atomic ions have a bound state at the threshold
when the nuclear charge attains its critical value. We shall also discuss a
necessary condition for four Coulomb charges (m1+, m2- , m3+ , m4- ), where
all masses are assumed finite, to form the stable system. The obtained
stability condition [2] is physical and is expressed through the required
minimal ratio of Jacobi masses. In particular, this proves that the hydrogenantihydrogen is unstable.
[1] D.K. Gridnev, arXiv math-ph/0611075 (2006).
[2] D.K. Gridnev and Carsten Greiner, Phys. Rev. Lett. 94, 223402 (2005).
Speaker
Sergey Levin, V. S. Buslaev, V. A. Fock Institute for Physics, St.
Title
“Asymptotic behavior of eigenfunctions of the Schrödinger operator for the
case of three one-dimensional particles”
We consider the simplest case of few-body scattering problem: the system
of three one-dimensional particles, interacting with only pair finite
potentials. Basing on analogies between the few-body scattering problem
and the problem of diffraction of the plane wave on the system of halftransparent screens, the asymptotic behavior of eigenfunctions, continuous
Petersburg University, Russia
Abstract
in all configuration space, has been derived. Some definite steps in this
direction have been done earlier in [1] Buslaev, V. S.; Merkur'ev, S. P.;
Salikov, S. P. Diffraction characteristics of scattering in a quantum system
of three one-dimensional particles. (Russian) Scattering theory. Theory of
oscillations (Russian), pp. 14--30, 183, Probl. Mat. Fiz., 9, Leningrad.
Univ., Leningrad, 1979.
Knowing the eigenfunctions asymptotic behavior, we reformulate the
initial three-body scattering problem into the problem of two-body
scattering on the potential with compact support. It should not appear any
principal obstacle for generalization of the method on the case of slowly
decreasing potential, for example Coulomb potential. Although in any
special case the procedure of generalization can not be done automatically
and require some additional efforts.
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Joseph H. Macek, University of Tennessee, Knoxville
“Feshbach resonances in atomic structure and dynamics”
Feshbach resonances are quasi-bound states that are represented, in first
approximation, as bound states in energetically closed channels that can
spontaneously break up into two or more fragments. I will review the
description of states that separate into two fragments within the framework
of multichannel effective range theory. Magnetic tuning of scattering
lengths is described within the context. Such tuning has opened up many
opportunities for fundamental studies of few-body systems. By using
multichannel zero-range pseudo-potentials it is possible to employ the
model when the few-body systems involve three or more fragments. The
emergence of Efimov states for critical values of the tuning parameters will
be discussed.
Keywords: Feshbach Resonances Psuedo-potentials
PACS: 34.50.-s
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Gheorghe Nenciu, Romanian Academy, University of Bucharest
“Exponential decay laws in perturbation theory of threshold and embedded
eigenvalues”
Exponential decay laws for the metastable states resulting from
perturbation of unstable eigenvalues are discussed. Eigenvalues embedded
in the continuum as well as threshold eigenvalues are considered. No
analytic continuation of the resolvent is required. The main result is about
threshold case: for Schrödinger operators in odd dimensions the leading
term of the life-time in the perturbation strength, ε, is of order εν/2 where ν
is an odd integer, ν ≥ 3. Examples covering all values of ν are given. For
eigenvalues embedded in the continuum the results sharpen the previous
ones.
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Maxim Olshanii, University of Southern California, Los Angeles
“Empirical manifestations of integrability in cold quantum gases”
Integrable quantum many-body systems traditionally belong to the domain
of mathematical physics, with little or no connection to experiments.
However, the experiments on confined quantum-degenerate gases have
recently yielded faithful realizations of a number of integrable systems,
thus making them phenomenologically relevant.
First, we illustrate the high predictive power of the integrable models for
the atomic experiments in several examples. These examples cover recent
experimental studies of the equation of state and correlation functions of
the one-dimensional Bose gas.
We further demonstrate that the integrabilty is not only an efficient
predictive tool but a source of new physics on its own. We show that the
presence of few-body conserved quantities in a quantum system leads to
dramatic, initial-state-dependent discrepancy between the state of the
system after relaxation and the predictions of thermodynamics. Using the
newly introduced concept of constrained thermal equilibrium we study
quantitatively the effects of the memory of the initial conditions. In
particular we consider the prohibition of chemical processes, isochronicity
of the monopole excitations, and suppressed/modified thermaization as the
suggested empirical consequences of integrability. Some of them have
been already successfully verified in experiments.
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Stephanie M. Reimann-Wacker, Lund Institute of Technology, Sweden
“Finite quantal systems – from semiconductor quantum dots to cold atoms
in traps”
Many-body systems that are set rotating may form vortices, characterized
by rotating motion around a central cavity. This is familiar to us from
every-day life: you can observe vortices while stirring your coffee, or
watching a hurricane. In quantum physics, vortices are known to occur in
superconducting films and rotating bosonic He-4 or fermionic He-3 liquids,
and recently became a hot topic in the research on cold atoms in traps. Here
we show that the rotation of trapped particles with a repulsive interaction
may lead to vortex formation regardless of whether the particles are bosons
or fermions. The exact many-particle wave function provides evidence that
the mechanism is very similar in both cases.
We discuss the close relation between rotating BECs and quantum dots at
strong magnetic fields. The vortices can stick to particles to form
composite particles, but also occur without association to any particular
particle. In quantum dots we find off-electron vortices that are localized,
giving rise to charge deficiency or holes in the density, with rotating
currents around them. The vortex formation is observable in the energetics
of the system. "Giant vortices" may form in anharmonic potentials. Here,
the vortices accumulate at the trap center, leading to large cores in the
electron and current densities.
Turning from single traps to periodic lattices, we comment upon the
analogies between optical lattices with cold fermionic atoms, and regular
arrays of few-electron quantum dots. Trapping a few (N < 12) fermions in
each of the single minima of the lattice, we find that the shell structure in
the quantum wells determines the magnetism, leading to a systematic
sequence of non-magnetic, ferromagnetic and antiferromagnetic states.
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Herbert Spohn, Technische Universität, München
“Fröhlich polarons and bipolarons, a mathematical physics perspective”
Over the past years there has been substantial progress on polaron-like
models. In my contribution I review results on the polaron, in particular
dispersion relation and strong coupling limit, and discuss recent results on
the phase diagram of the bipolaron.
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Thomas Østergaard Sørensen, Aalborg University, Denmark
“Regularity of eigenfunctions and one-electron densities of Coulombic
Schrödinger operators”
We study the regularity of molecular eigenfunctions near the singularities
of the many-body Coulomb potential. The results obtained have been used
to prove that the corresponding electron density is real analytic away from
the positions of the nuclei. They are also essential ingredients in studying
the density in the vicinity of the nuclei; we prove the existence of the third
derivative of the spherical average of the density at a nucleus, and discuss
its sign. We also discuss positivity and decay of the spherical average.
Abstract
This is joint work with S. Fournais (University of Aarhus & CNRS), and
M. and T. Hoffmann-Ostenhof (Vienna).
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Martin Thøgersen, Department of Physics and Astronomy, AU
“Spatial correlations in Bose gases”
We consider an N-boson system with two-body interactions in an external
harmonic trap. Correlations are introduced directly in the wavefunction, the
energy is minimized stochastically, and the condensate fraction is obtained.
The two-body interaction is characterized by the s-wave scattering length,
a, and both the weakly and strongly interacting regimes are investigated.