107 research outputs found
A note on the orthogonality equation with two functions
The aim of this paper is to describe the solution (f, g) of the equation
f(x)|g(y) = x|y , x,y∈ D, where f, g : D → Y , X, Y are Hilbert spaces over the same field K ∈ {R,C}, D is a dense subspace of X
An example of spectral phase transition phenomenon in a class of Jacobi matrices with periodically modulated weights
We consider self-adjoint unbounded Jacobi matrices with diagonal q_n=n and
weights \lambda_n=c_n n, where c_n is a 2-periodical sequence of real numbers.
The parameter space is decomposed into several separate regions, where the
spectrum is either purely absolutely continuous or discrete. This constitutes
an example of the spectral phase transition of the first order. We study the
lines where the spectral phase transition occurs, obtaining the following main
result: either the interval (-\infty;1/2) or the interval (1/2;+\infty) is
covered by the absolutely continuous spectrum, the remainder of the spectrum
being pure point. The proof is based on finding asymptotics of generalized
eigenvectors via the Birkhoff-Adams Theorem. We also consider the degenerate
case, which constitutes yet another example of the spectral phase transition
On the Two Spectra Inverse Problem for Semi-Infinite Jacobi Matrices
We present results on the unique reconstruction of a semi-infinite Jacobi
operator from the spectra of the operator with two different boundary
conditions. This is the discrete analogue of the Borg-Marchenko theorem for
Schr{\"o}dinger operators in the half-line. Furthermore, we give necessary and
sufficient conditions for two real sequences to be the spectra of a Jacobi
operator with different boundary conditions.Comment: In this slightly revised version we have reworded some of the
theorems, and we updated two reference
The Two-Spectra Inverse Problem for Semi-Infinite Jacobi Matrices in The Limit-Circle Case
We present a technique for reconstructing a semi-infinite Jacobi operator in
the limit circle case from the spectra of two different self-adjoint
extensions. Moreover, we give necessary and sufficient conditions for two real
sequences to be the spectra of two different self-adjoint extensions of a
Jacobi operator in the limit circle case.Comment: 26 pages. Changes in the presentation of some result
An expansion for polynomials orthogonal over an analytic Jordan curve
We consider polynomials that are orthogonal over an analytic Jordan curve L
with respect to a positive analytic weight, and show that each such polynomial
of sufficiently large degree can be expanded in a series of certain integral
transforms that converges uniformly in the whole complex plane. This expansion
yields, in particular and simultaneously, Szego's classical strong asymptotic
formula and a new integral representation for the polynomials inside L. We
further exploit such a representation to derive finer asymptotic results for
weights having finitely many singularities (all of algebraic type) on a thin
neighborhood of the orthogonality curve. Our results are a generalization of
those previously obtained in [7] for the case of L being the unit circle.Comment: 15 pages, 1 figur
One-sided Cauchy-Stieltjes Kernel Families
This paper continues the study of a kernel family which uses the
Cauchy-Stieltjes kernel in place of the celebrated exponential kernel of the
exponential families theory. We extend the theory to cover generating measures
with support that is unbounded on one side. We illustrate the need for such an
extension by showing that cubic pseudo-variance functions correspond to
free-infinitely divisible laws without the first moment. We also determine the
domain of means, advancing the understanding of Cauchy-Stieltjes kernel
families also for compactly supported generating measures
The smallest eigenvalue of Hankel matrices
Let H_N=(s_{n+m}),n,m\le N denote the Hankel matrix of moments of a positive
measure with moments of any order. We study the large N behaviour of the
smallest eigenvalue lambda_N of H_N. It is proved that lambda_N has exponential
decay to zero for any measure with compact support. For general determinate
moment problems the decay to 0 of lambda_N can be arbitrarily slow or
arbitrarily fast. In the indeterminate case, where lambda_N is known to be
bounded below by a positive constant, we prove that the limit of the n'th
smallest eigenvalue of H_N for N tending to infinity tends rapidly to infinity
with n. The special case of the Stieltjes-Wigert polynomials is discussed
Tunneling times with covariant measurements
We consider the time delay of massive, non-relativistic, one-dimensional
particles due to a tunneling potential. In this setting the well-known Hartman
effect asserts that often the sub-ensemble of particles going through the
tunnel seems to cross the tunnel region instantaneously. An obstacle to the
utilization of this effect for getting faster signals is the exponential
damping by the tunnel, so there seems to be a trade-off between speedup and
intensity. In this paper we prove that this trade-off is never in favor of
faster signals: the probability for a signal to reach its destination before
some deadline is always reduced by the tunnel, for arbitrary incoming states,
arbitrary positive and compactly supported tunnel potentials, and arbitrary
detectors. More specifically, we show this for several different ways to define
``the same incoming state'' and ''the same detector'' when comparing the
settings with and without tunnel potential. The arrival time measurements are
expressed in the time-covariant approach, but we also allow the detection to be
a localization measurement at a later time.Comment: 12 pages, 2 figure
Time ordering and counting statistics
The basic quantum mechanical relation between fluctuations of transported
charge and current correlators is discussed. It is found that, as a rule, the
correlators are to be time-ordered in an unusual way. Instances where the
difference with the conventional ordering matters are illustrated by means of a
simple scattering model. We apply the results to resolve a discrepancy
concerning the third cumulant of charge transport across a quantum point
contact.Comment: 19 pages, 1 figure; inconsequential mistake and typos correcte
Free motion time-of-arrival operator and probability distribution
We reappraise and clarify the contradictory statements found in the
literature concerning the time-of-arrival operator introduced by Aharonov and
Bohm in Phys. Rev. {\bf 122}, 1649 (1961). We use Naimark's dilation theorem to
reproduce the generalized decomposition of unity (or POVM) from any
self-adjoint extension of the operator, emphasizing a natural one, which arises
from the analogy with the momentum operator on the half-line. General time
operators are set within a unifying perspective. It is shown that they are not
in general related to the time of arrival, even though they may have the same
form.Comment: 10 a4 pages, no figure
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