331 research outputs found
Low rank perturbations and the spectral statistics of pseudointegrable billiards
We present an efficient method to solve Schr\"odinger's equation for
perturbations of low rank. In particular, the method allows to calculate the
level counting function with very little numerical effort. To illustrate the
power of the method, we calculate the number variance for two pseudointegrable
quantum billiards: the barrier billiard and the right triangle billiard
(smallest angle ). In this way, we obtain precise estimates for the
level compressibility in the semiclassical (high energy) limit. In both cases,
our results confirm recent theoretical predictions, based on periodic orbit
summation.Comment: 4 page
Collective versus Single--Particle Motion in Quantum Many--Body Systems: Spreading and its Semiclassical Interpretation
We study the interplay between collective and incoherent single-particle
motion in a model of two chains of particles whose interaction comprises a
non-integrable part. In the perturbative regime, but for a general form of the
interaction, we calculate the spectral density for collective excitations. We
obtain the remarkable result that it always has a unique semiclassical
interpretation. We show this by a proper renormalization procedure which allows
us to map our system to a Caldeira-Leggett--type of model in which the bath is
part of the system.Comment: 4 page
Ordering of Energy Levels in Heisenberg Models and Applications
In a recent paper we conjectured that for ferromagnetic Heisenberg models the
smallest eigenvalues in the invariant subspaces of fixed total spin are
monotone decreasing as a function of the total spin and called this property
ferromagnetic ordering of energy levels (FOEL). We have proved this conjecture
for the Heisenberg model with arbitrary spins and coupling constants on a
chain. In this paper we give a pedagogical introduction to this result and also
discuss some extensions and implications. The latter include the property that
the relaxation time of symmetric simple exclusion processes on a graph for
which FOEL can be proved, equals the relaxation time of a random walk on the
same graph. This equality of relaxation times is known as Aldous' Conjecture.Comment: 20 pages, contribution for the proceedings of QMATH9, Giens,
September 200
Can billiard eigenstates be approximated by superpositions of plane waves?
The plane wave decomposition method (PWDM) is one of the most popular
strategies for numerical solution of the quantum billiard problem. The method
is based on the assumption that each eigenstate in a billiard can be
approximated by a superposition of plane waves at a given energy. By the
classical results on the theory of differential operators this can indeed be
justified for billiards in convex domains. On the contrary, in the present work
we demonstrate that eigenstates of non-convex billiards, in general, cannot be
approximated by any solution of the Helmholtz equation regular everywhere in
(in particular, by linear combinations of a finite number of plane waves
having the same energy). From this we infer that PWDM cannot be applied to
billiards in non-convex domains. Furthermore, it follows from our results that
unlike the properties of integrable billiards, where each eigenstate can be
extended into the billiard exterior as a regular solution of the Helmholtz
equation, the eigenstates of non-convex billiards, in general, do not admit
such an extension.Comment: 23 pages, 5 figure
Correlations between spectra with different symmetry: any chance to be observed?
A standard assumption in quantum chaology is the absence of correlation
between spectra pertaining to different symmetries. Doubts were raised about
this statement for several reasons, in particular, because in semiclassics
spectra of different symmetry are expressed in terms of the same set of
periodic orbits. We reexamine this question and find absence of correlation in
the universal regime. In the case of continuous symmetry the problem is reduced
to parametric correlation, and we expect correlations to be present up to a
certain time which is essentially classical but larger than the ballistic time
Ölümünün 50 nci yıldönümü münasebetile Namık Kemal
Taha Toros Arşivi, Dosya Adı: Namık Kemalİstanbul Kalkınma Ajansı (TR10/14/YEN/0033) İstanbul Development Agency (TR10/14/YEN/0033
Expanded boundary integral method and chaotic time-reversal doublets in quantum billiards
We present the expanded boundary integral method for solving the planar
Helmholtz problem, which combines the ideas of the boundary integral method and
the scaling method and is applicable to arbitrary shapes. We apply the method
to a chaotic billiard with unidirectional transport, where we demonstrate
existence of doublets of chaotic eigenstates, which are quasi-degenerate due to
time-reversal symmetry, and a very particular level spacing distribution that
attains a chaotic Shnirelman peak at short energy ranges and exhibits GUE-like
statistics for large energy ranges. We show that, as a consequence of such
particular level statistics or algebraic tunneling between disjoint chaotic
components connected by time-reversal operation, the system exhibits quantum
current reversals.Comment: 18 pages, 8 figures, with 3 additional GIF animations available at
http://chaos.fiz.uni-lj.si/~veble/boundary
Tunable Lyapunov exponent in inverse magnetic billiards
The stability properties of the classical trajectories of charged particles
are investigated in a two dimensional stadium-shaped inverse magnetic domain,
where the magnetic field is zero inside the stadium domain and constant
outside. In the case of infinite magnetic field the dynamics of the system is
the same as in the Bunimovich billiard, i.e., ergodic and mixing. However, for
weaker magnetic fields the phase space becomes mixed and the chaotic part
gradually shrinks. The numerical measurements of the Lyapunov exponent
(performed with a novel method) and the integrable/chaotic phase space volume
ratio show that both quantities can be smoothly tuned by varying the external
magnetic field. A possible experimental realization of the arrangement is also
discussed.Comment: 4 pages, 6 figure
Level spacing distribution of pseudointegrable billiard
In this paper, we examine the level spacing distribution of the
rectangular billiard with a single point-like scatterer, which is known as
pseudointegrable. It is shown that the observed is a new type, which is
quite different from the previous conclusion. Even in the strong coupling
limit, the Poisson-like behavior rather than Wigner-like is seen for ,
although the level repulsion still remains in the small region. The
difference from the previous works is analyzed in detail.Comment: 11 pages, REVTeX file, 3 PostScript Figure
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