892,818 research outputs found
Quantal Andreev billiards: Semiclassical approach to mesoscale oscillations in the density of states
Andreev billiards are finite, arbitrarily-shaped, normal-state regions,
surrounded by superconductor. At energies below the superconducting gap,
single-quasiparticle excitations are confined to the normal region and its
vicinity, the essential mechanism for this confinement being Andreev
reflection. This Paper develops and implements a theoretical framework for the
investigation of the short-wave quantal properties of these
single-quasiparticle excitations. The focus is primarily on the relationship
between the quasiparticle energy eigenvalue spectrum and the geometrical shape
of the normal-state region, i.e., the question of spectral geometry in the
novel setting of excitations confined by a superconducting pair-potential.
Among the central results of this investigation are two semiclassical trace
formulas for the density of states. The first, a lower-resolution formula,
corresponds to the well-known quasiclassical approximation, conventionally
invoked in settings involving superconductivity. The second, a
higher-resolution formula, allows the density of states to be expressed in
terms of: (i) An explicit formula for the level density, valid in the
short-wave limit, for billiards of arbitrary shape and dimensionality. This
level density depends on the billiard shape only through the set of
stationary-length chords of the billiard and the curvature of the boundary at
the endpoints of these chords; and (ii) Higher-resolution corrections to the
level density, expressed as a sum over periodic orbits that creep around the
billiard boundary. Owing to the fact that these creeping orbits are much longer
than the stationary chords, one can, inter alia, hear the stationary chords of
Andreev billiards.Comment: 52 pages, 15 figures, 1 table, RevTe
Clarke subgradients of stratifiable functions
We establish the following result: if the graph of a (nonsmooth)
real-extended-valued function
is closed and admits a Whitney stratification, then the norm of the gradient of
at relative to the stratum containing bounds from below
all norms of Clarke subgradients of at . As a consequence, we obtain
some Morse-Sard type theorems as well as a nonsmooth Kurdyka-\L ojasiewicz
inequality for functions definable in an arbitrary o-minimal structure
Quantum Integrable Model of an Arrangement of Hyperplanes
The goal of this paper is to give a geometric construction of the Bethe
algebra (of Hamiltonians) of a Gaudin model associated to a simple Lie algebra.
More precisely, in this paper a quantum integrable model is assigned to a
weighted arrangement of affine hyperplanes. We show (under certain assumptions)
that the algebra of Hamiltonians of the model is isomorphic to the algebra of
functions on the critical set of the corresponding master function. For a
discriminantal arrangement we show (under certain assumptions) that the
symmetric part of the algebra of Hamiltonians is isomorphic to the Bethe
algebra of the corresponding Gaudin model. It is expected that this
correspondence holds in general (without the assumptions). As a byproduct of
constructions we show that in a Gaudin model (associated to an arbitrary simple
Lie algebra), the Bethe vector, corresponding to an isolated critical point of
the master function, is nonzero
Special K\"ahler-Ricci potentials on compact K\"ahler manifolds
A special K\"ahler-Ricci potential on a K\"ahler manifold is any nonconstant
function such that is a Killing vector field
and, at every point with , all nonzero tangent vectors orthogonal
to and are eigenvectors of both and
the Ricci tensor. For instance, this is always the case if is a
nonconstant function on a K\"ahler manifold of complex
dimension and the metric , defined wherever , is Einstein. (When such exists, may be called {\it
almost-everywhere conformally Einstein}.) We provide a complete classification
of compact K\"ahler manifolds with special K\"ahler-Ricci potentials and use it
to prove a structure theorem for compact K\"ahler manifolds of any complex
dimension which are almost-everywhere conformally Einstein.Comment: 45 pages, AMSTeX, submitted to Journal f\"ur die reine und angewandte
Mathemati
Converse growth estimates for ODEs with slowly growing solutions
Let be linearly independent solutions of , where the
coefficient is an analytic function in the open unit disc of
. It is shown that many properties of this differential equation
can be described in terms of the subharmonic auxiliary function . For example, the case when and is normal, is characterized by the
condition . Different
types of Blaschke-oscillatory equations are also described in terms of harmonic
majorants of .
Even if are bounded linearly independent solutions of ,
it is possible that or
is non-normal. These results relate to sharpness discussion of recent
results in the literature, and are succeeded by a detailed analysis of
differential equations with bounded solutions. Analogues results for the
Nevanlinna class are also considered, by taking advantage of Nevanlinna
interpolating sequences.
It is shown that, instead of considering solutions with prescribed zeros, it
is possible to construct a bounded solution of in such a way that it
solves an interpolation problem natural to bounded analytic functions, while
remains to be a Carleson measure.Comment: 29 page
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