10.00-10.40
S.T. Kuroda: Some topics in eigenvalue computations
10.45-11.15
I. Stoll: The life and work of Christian Doppler
11.20-12.00
E.H. Lieb: The dilute, cold Bose gas: a truly quantum-mechanical
many-body problem
12.10-14.00 Lunch break
Chair: L. Hlavaty
14.00-14.40
J.-M. Combes: Edge conductivity in quantum Hall systems
14.45-15.25
P. Exner: Resonance effects in leaky quantum wires
15.30-16.00
Coffe break
Chair: J. Dittrich
16.00-16.40
A. Joye: Adiabatic approximations and exponential asymptotics
16.45-17.15
M. Znojil: Magyari equations as a non-square eigenvalue problem
Abstracts:
J.E. Avron: Euler disc
Euler disc is a mechanical toy with intriguing behavior that is not full
understood. The talk will consist of demonstrations and an outline of the
theory. Some open problems shall also be discussed.
J.-M. Combes: Edge conductivity
in quantum Hall systems
We discuss quantization of edge conductivity in a one electron model confined
to a semi-infinite planar domain containing impurities. A sum rule allows
to derive exact quantization in various situations. In particular it is shown
that deviation of edge conductivity in the Nth Landau band from its ideal
value N is linked to existence of "edge currents without edges". Exact quantization
is also shown to hold in some models of high disorder.
H.-D. Doebner: Extensions
of quantum mechanics - nonlinear Schroedinger equations
A family of nonlinear extensions of nonrelativistic quantum mechanical
evolution equations is presented. The family is based on on a quantisation
method for the kinematics (Quantum Borel Kinematic). A corresponding time
dependence yields a family of nonlinear Schrödinger equations. Difficulties
of nonlinear operators and nonlinear evolutions in the usual quamtum mechanical
framework are discussed. Examples to measure such nonlineraties through quantum
mechanical precision experiments are explained.
P. Exner: Resonance effects
in leakyquantum wires
We consider a model of a "leaky" the Hamiltonian of which is a two-dimensional
Schroedinger operator, and ask about resonance effects for negative-energy
states guided along such a structure. Using an approximation by point interactions
we present a numerical evidence that these systems may exhibit two types
of resonances, due to propagating-mode reflections and due to quantum tunelling.
We also analyse a solvable model of a straight wire and a quantum dot modeled
by a point.
A. Joye: Adiabatic approximations
and exponential asymptotics
We discuss the quantum adiabatic approximation and describe recent work
with G. Hagedorn that determines the time development of exponentially small
non-adiabatic transitions for some special models.
S.T. Kuroda: Some topics
in eigenvalue computations
We review our works (works done in our group) on this topics. We will
talk about, possibly not standard, methods of computing eigenvalues of, say,
Schroedinger operators. The emphasis will be on functional-analytic investigations,
not on acutual computations.
E.H. Lieb: The dilute, cold
Bose gas: a truly quantum-mechanical many-body problem
The peculiar quantum-mechanical properties of the ground state of Bose
gases that have been proved rigorously in the last few years (with R. Seiringer,
J-P. Solovej and J. Yngvason) will be reviewed. For the low density gas with
finite range interactions these properties include the leading order term
in the ground state energy, the validity of the Gross-Pitaevskii description
in traps, Bose-Einstein condensationand superfluidity in traps, and the transition
from 3-dimensional behaviorto 1-dimensional behavior as the cross-section
of the trap decreases. The latter is a highly quantum-mechanical phenomenon.
For the charged Bose gas at high density, the leading term in the energy
found by Foldy in 1961 for the one-component gas and Dyson's conjecture of
the N^{7/5} law for the two-component gas has also been verified. These results
help justify Bogolubov's 1947 theory of pairing in Bose gases.
P. Seba: Random matrix theory
and statistical properties of exotic systems
We will demonstrate that statistical distributions originally coming from
the random matrix theory can be used to describe properties of such exotic
systems like bus transport in Mexico, distribution of cars on German highways,
cross channel correlations of human brain potentials (EEG) and distribution
of phonems in various languages.
I. Stoll: The life and work
of Christian Doppler
Short synopsis: Christian Doppler's youth and ambitions. Doppler's teaching
activities in Prague. Membership of Royal Bohemian Society of Sciences. The
highlight of Dopples's scientific achievements. The retreat in Banska Stiavnica
and variations of earth magnetism. Last years in Vienna; at the beginnings
of Austrian physics. The journey of Doppler's vision into physics (Fizeau,
Huggins, Mach, Einstein).
J. Tolar: Ten years of the
Doppler Institute
The Faculty of Nuclear Sciences and Physical Engineering of Czech Technical
University in Prague concentrates on engineering branches requiring solid
knowledge of mathematics and physics. Our Seminar of Mathematical Physics,
since its start in 1988, has become center of our cooperation on which the
Doppler Institute of mathematical physics (DI) was founded in 1993 as a research
and pedagogical division of the Faculty. The objective of the scientific
research of the DI collaborators is to develop core disciplines of modern
mathematical quantum physics. In its pedagogical activities DI provides
help to students at the beginning of their active scientific career. This
contribution will be devoted to the survey of our scientific and pedagogical
activities during the past 10 years as well as to our plans for the future.
M. Znojil: Magyari equations
as a non-square eigenvalue problem
The standard formulation of the linear N by N problem Hx = Ex will be
extended to certain non-square Hamiltonians H in a "feasible" non-linear
manner. A few possible applications will be mentioned though, mainly, our
attention will be paid to the Magyari's project of the determination of a
"quasi-harmonic" degeneracy of a general polynomial oscillator. On mathematical
side we will outline a generalized Rayleigh-Schroedinger perturbation construction
of solutions. In particular, its (so called 1/D - expansion) application
to polynomial oscillators will be described in detail, with emphasis on the
enigmatic exact solvability of this problem in zero order (where two alternative
constructions are known up to now).
