2,909 research outputs found
Stability of the magnetic Schr\"odinger operator in a waveguide
The spectrum of the Schr\"odinger operator in a quantum waveguide is known to
be unstable in two and three dimensions. Any enlargement of the waveguide
produces eigenvalues beneath the continuous spectrum. Also if the waveguide is
bent eigenvalues will arise below the continuous spectrum. In this paper a
magnetic field is added into the system. The spectrum of the magnetic
Schr\"odinger operator is proved to be stable under small local deformations
and also under small bending of the waveguide. The proof includes a magnetic
Hardy-type inequality in the waveguide, which is interesting in its own
Braids of entangled particle trajectories
In many applications, the two-dimensional trajectories of fluid particles are
available, but little is known about the underlying flow. Oceanic floats are a
clear example. To extract quantitative information from such data, one can
measure single-particle dispersion coefficients, but this only uses one
trajectory at a time, so much of the information on relative motion is lost. In
some circumstances the trajectories happen to remain close long enough to
measure finite-time Lyapunov exponents, but this is rare. We propose to use
tools from braid theory and the topology of surface mappings to approximate the
topological entropy of the underlying flow. The procedure uses all the
trajectory data and is inherently global. The topological entropy is a measure
of the entanglement of the trajectories, and converges to zero if they are not
entangled in a complex manner (for instance, if the trajectories are all in a
large vortex). We illustrate the techniques on some simple dynamical systems
and on float data from the Labrador sea.Comment: 24 pages, 21 figures. PDFLaTeX with RevTeX4 macros. Matlab code
included with source. Fixed an inconsistent convention problem. Final versio
Strong contraction of the representations of the three dimensional Lie algebras
For any Inonu-Wigner contraction of a three dimensional Lie algebra we
construct the corresponding contractions of representations. Our method is
quite canonical in the sense that in all cases we deal with realizations of the
representations on some spaces of functions; we contract the differential
operators on those spaces along with the representation spaces themselves by
taking certain pointwise limit of functions. We call such contractions strong
contractions. We show that this pointwise limit gives rise to a direct limit
space. Many of these contractions are new and in other examples we give a
different proof
Abelian subgroups of Garside groups
In this paper, we show that for every abelian subgroup of a Garside
group, some conjugate consists of ultra summit elements and the
centralizer of is a finite index subgroup of the normalizer of .
Combining with the results on translation numbers in Garside groups, we obtain
an easy proof of the algebraic flat torus theorem for Garside groups and solve
several algorithmic problems concerning abelian subgroups of Garside groups.Comment: This article replaces our earlier preprint "Stable super summit sets
in Garside groups", arXiv:math.GT/060258
Dual generators of the fundamental group and the moduli space of flat connections
We define the dual of a set of generators of the fundamental group of an
oriented two-surface of genus with punctures and the
associated surface with a disc removed. This dual is
another set of generators related to the original generators via an involution
and has the properties of a dual graph. In particular, it provides an algebraic
prescription for determining the intersection points of a curve representing a
general element of the fundamental group with the
representatives of the generators and the order in which these intersection
points occur on the generators.We apply this dual to the moduli space of flat
connections on and show that when expressed in terms both, the
holonomies along a set of generators and their duals, the Poisson structure on
the moduli space takes a particularly simple form. Using this description of
the Poisson structure, we derive explicit expressions for the Poisson brackets
of general Wilson loop observables associated to closed, embedded curves on the
surface and determine the associated flows on phase space. We demonstrate that
the observables constructed from the pairing in the Chern-Simons action
generate of infinitesimal Dehn twists and show that the mapping class group
acts by Poisson isomorphisms.Comment: 54 pages, 13 .eps figure
Exact Casimir Interaction Between Semitransparent Spheres and Cylinders
A multiple scattering formulation is used to calculate the force, arising
from fluctuating scalar fields, between distinct bodies described by
-function potentials, so-called semitransparent bodies. (In the limit
of strong coupling, a semitransparent boundary becomes a Dirichlet one.) We
obtain expressions for the Casimir energies between disjoint parallel
semitransparent cylinders and between disjoint semitransparent spheres. In the
limit of weak coupling, we derive power series expansions for the energy, which
can be exactly summed, so that explicit, very simple, closed-form expressions
are obtained in both cases. The proximity force theorem holds when the objects
are almost touching, but is subject to large corrections as the bodies are
moved further apart.Comment: 5 pages, 4 eps figures; expanded discussion of previous work and
additional references added, minor typos correcte
The BCS gap equation for spin-polarized fermions
We study the BCS gap equation for a Fermi gas with unequal population of
spin-up and spin-down states. For , with the
temperature and the chemical potential difference, the question of
existence of non-trivial solutions can be reduced to spectral properties of a
linear operator, similar to the unpolarized case studied previously in
\cite{FHNS,HHSS,HS}. For the phase diagram is more
complicated, however. We derive upper and lower bounds for the critical
temperature, and study their behavior in the small coupling limit.Comment: 23 pages, 1 figur
The spectral shift function and spectral flow
This paper extends Krein's spectral shift function theory to the setting of
semifinite spectral triples. We define the spectral shift function under these
hypotheses via Birman-Solomyak spectral averaging formula and show that it
computes spectral flow.Comment: 47 page
Scattering through a straight quantum waveguide with combined boundary conditions
Scattering through a straight two-dimensional quantum waveguide Rx(0,d) with
Dirichlet boundary conditions on (-\infty,0)x{y=0} \cup (0,\infty)x{y=d} and
Neumann boundary condition on (-infty,0)x{y=d} \cup (0,\infty)x{y=0} is
considered using stationary scattering theory. The existence of a matching
conditions solution at x=0 is proved. The use of stationary scattering theory
is justified showing its relation to the wave packets motion. As an
illustration, the matching conditions are also solved numerically and the
transition probabilities are shown.Comment: 26 pages, 3 figure
Quantum Versus Mean Field Behavior of Normal Modes of a Bose-Einstein Condensate in a Magnetic Trap
Quantum evolution of a collective mode of a Bose-Einstein condensate
containing a finite number N of particles shows the phenomena of collapses and
revivals. The characteristic collapse time depends on the scattering length,
the initial amplitude of the mode and N. The corresponding time values have
been derived analytically under certain approximation and numerically for the
parabolic atomic trap. The revival of the mode at time of several seconds, as a
direct evidence of the effect, can occur, if the normal component is
significantly suppressed.
We also discuss alternative means to verify the proposed mechanism.Comment: minor corrections are introduced into the tex
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