29,265 research outputs found
Birman-Schwinger and the number of Andreev states in BCS superconductors
The number of bound states due to inhomogeneities in a BCS superconductor is
usually established either by variational means or via exact solutions of
particularly simple, symmetric perturbations. Here we propose estimating the
number of sub-gap states using the Birman-Schwinger principle. We show how to
obtain upper bounds on the number of sub-gap states for small normal regions
and derive a suitable Cwikel-Lieb-Rozenblum inequality. We also estimate the
number of such states for large normal regions using high dimensional
generalizations of the Szego theorem. The method works equally well for local
inhomogeneities of the order parameter and for external potentials.Comment: Final version to appear in Phys Rev
Perturbation Theory of Schr\"odinger Operators in Infinitely Many Coupling Parameters
In this paper we study the behavior of Hamilton operators and their spectra
which depend on infinitely many coupling parameters or, more generally,
parameters taking values in some Banach space. One of the physical models which
motivate this framework is a quantum particle moving in a more or less
disordered medium. One may however also envisage other scenarios where
operators are allowed to depend on interaction terms in a manner we are going
to discuss below. The central idea is to vary the occurring infinitely many
perturbing potentials independently. As a side aspect this then leads naturally
to the analysis of a couple of interesting questions of a more or less purely
mathematical flavor which belong to the field of infinite dimensional
holomorphy or holomorphy in Banach spaces. In this general setting we study in
particular the stability of selfadjointness of the operators under discussion
and the analyticity of eigenvalues under the condition that the perturbing
potentials belong to certain classes.Comment: 25 pages, Late
Analytic structure of Bloch functions for linear molecular chains
This paper deals with Hamiltonians of the form H=-{\bf \nabla}^2+v(\rr),
with v(\rr) periodic along the direction, . The
wavefunctions of are the well known Bloch functions
\psi_{n,\lambda}(\rr), with the fundamental property
and
. We give the generic analytic structure
(i.e. the Riemann surface) of \psi_{n,\lambda}(\rr) and their corresponding
energy, , as functions of . We show that
and are different branches of two multi-valued
analytic functions, and , with an essential
singularity at and additional branch points, which are generically
of order 1 and 3, respectively. We show where these branch points come from,
how they move when we change the potential and how to estimate their location.
Based on these results, we give two applications: a compact expression of the
Green's function and a discussion of the asymptotic behavior of the density
matrix for insulating molecular chains.Comment: 13 pages, 11 figure
Quantum Inverse Square Interaction
Hamiltonians with inverse square interaction potential occur in the study of
a variety of physical systems and exhibit a rich mathematical structure. In
this talk we briefly mention some of the applications of such Hamiltonians and
then analyze the case of the N-body rational Calogero model as an example. This
model has recently been shown to admit novel solutions, whose properties are
discussed.Comment: Talk presented at the conference "Space-time and Fundamental
Interactions: Quantum Aspects" in honour of Prof. A.P.Balachandran's 65th
birthday, Vietri sul Mare, Italy, 26 - 31 May, 2003, Latex file, 9 pages.
Some references added in the replaced versio
Sharp eigenvalue enclosures for the perturbed angular Kerr-Newman Dirac operator
A certified strategy for determining sharp intervals of enclosure for the
eigenvalues of matrix differential operators with singular coefficients is
examined. The strategy relies on computing the second order spectrum relative
to subspaces of continuous piecewise linear functions. For smooth perturbations
of the angular Kerr-Newman Dirac operator, explicit rates of convergence due to
regularity of the eigenfunctions are established. Existing benchmarks are
validated and sharpened by several orders of magnitude in the unperturbed
setting.Comment: 27 pages, 2 figures, 5 tables. Some errors fixe
Resonances Width in Crossed Electric and Magnetic Fields
We study the spectral properties of a charged particle confined to a
two-dimensional plane and submitted to homogeneous magnetic and electric fields
and an impurity potential. We use the method of complex translations to prove
that the life-times of resonances induced by the presence of electric field are
at least Gaussian long as the electric field tends to zero.Comment: 3 figure
Analysis of unbounded operators and random motion
We study infinite weighted graphs with view to \textquotedblleft limits at
infinity,\textquotedblright or boundaries at infinity. Examples of such
weighted graphs arise in infinite (in practice, that means \textquotedblleft
very\textquotedblright large) networks of resistors, or in statistical
mechanics models for classical or quantum systems. But more generally our
analysis includes reproducing kernel Hilbert spaces and associated operators on
them. If is some infinite set of vertices or nodes, in applications the
essential ingredient going into the definition is a reproducing kernel Hilbert
space; it measures the differences of functions on evaluated on pairs of
points in . And the Hilbert norm-squared in will represent
a suitable measure of energy. Associated unbounded operators will define a
notion or dissipation, it can be a graph Laplacian, or a more abstract
unbounded Hermitian operator defined from the reproducing kernel Hilbert space
under study. We prove that there are two closed subspaces in reproducing kernel
Hilbert space which measure quantitative notions of limits at
infinity in , one generalizes finite-energy harmonic functions in
, and the other a deficiency index of a natural operator in
associated directly with the diffusion. We establish these
results in the abstract, and we offer examples and applications. Our results
are related to, but different from, potential theoretic notions of
\textquotedblleft boundaries\textquotedblright in more standard random walk
models. Comparisons are made.Comment: 38 pages, 4 tables, 3 figure
Quantum graphs with singular two-particle interactions
We construct quantum models of two particles on a compact metric graph with
singular two-particle interactions. The Hamiltonians are self-adjoint
realisations of Laplacians acting on functions defined on pairs of edges in
such a way that the interaction is provided by boundary conditions. In order to
find such Hamiltonians closed and semi-bounded quadratic forms are constructed,
from which the associated self-adjoint operators are extracted. We provide a
general characterisation of such operators and, furthermore, produce certain
classes of examples. We then consider identical particles and project to the
bosonic and fermionic subspaces. Finally, we show that the operators possess
purely discrete spectra and that the eigenvalues are distributed following an
appropriate Weyl asymptotic law
Scaling limits of integrable quantum field theories
Short distance scaling limits of a class of integrable models on
two-dimensional Minkowski space are considered in the algebraic framework of
quantum field theory. Making use of the wedge-local quantum fields generating
these models, it is shown that massless scaling limit theories exist, and
decompose into (twisted) tensor products of chiral, translation-dilation
covariant field theories. On the subspace which is generated from the vacuum by
the observables localized in finite light ray intervals, this symmetry can be
extended to the M\"obius group. The structure of the interval-localized
algebras in the chiral models is discussed in two explicit examples.Comment: Revised version: erased typos, improved formulations, and corrections
of Lemma 4.8/Prop. 4.9. As published in RMP. 43 pages, 1 figur
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