2,490 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
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
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
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
Elliptic operators in even subspaces
In the paper we consider the theory of elliptic operators acting in subspaces
defined by pseudodifferential projections. This theory on closed manifolds is
connected with the theory of boundary value problems for operators violating
Atiyah-Bott condition. We prove an index formula for elliptic operators in
subspaces defined by even projections on odd-dimensional manifolds and for
boundary value problems, generalizing the classical result of Atiyah-Bott.
Besides a topological contribution of Atiyah-Singer type, the index formulas
contain an invariant of subspaces defined by even projections. This homotopy
invariant can be expressed in terms of the eta-invariant. The results also shed
new light on P.Gilkey's work on eta-invariants of even-order operators.Comment: 39 pages, 2 figure
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
Positive Measure Spectrum for Schroedinger Operators with Periodic Magnetic Fields
We study Schroedinger operators with periodic magnetic field in Euclidean
2-space, in the case of irrational magnetic flux. Positive measure Cantor
spectrum is generically expected in the presence of an electric potential. We
show that, even without electric potential, the spectrum has positive measure
if the magnetic field is a perturbation of a constant one.Comment: 17 page
Sufficient conditions for the existence of bound states in a central potential
We show how a large class of sufficient conditions for the existence of bound
states, in non-positive central potentials, can be constructed. These
sufficient conditions yield upper limits on the critical value,
, of the coupling constant (strength), , of the
potential, , for which a first -wave bound state appears.
These upper limits are significantly more stringent than hitherto known
results.Comment: 7 page
Upper and lower limits on the number of bound states in a central potential
In a recent paper new upper and lower limits were given, in the context of
the Schr\"{o}dinger or Klein-Gordon equations, for the number of S-wave
bound states possessed by a monotonically nondecreasing central potential
vanishing at infinity. In this paper these results are extended to the number
of bound states for the -th partial wave, and results are also
obtained for potentials that are not monotonic and even somewhere positive. New
results are also obtained for the case treated previously, including the
remarkably neat \textit{lower} limit with (valid in the Schr\"{o}dinger case, for a class of potentials
that includes the monotonically nondecreasing ones), entailing the following
\textit{lower} limit for the total number of bound states possessed by a
monotonically nondecreasing central potential vanishing at infinity: N\geq
\{\{(\sigma+1)/2\}\} {(\sigma+3)/2\} \}/2 (here the double braces denote of
course the integer part).Comment: 44 pages, 5 figure
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