5 research outputs found
Resonances for "large" ergodic systems in one dimension: a review
The present note reviews recent results on resonances for one-dimensional
quantum ergodic systems constrained to a large box. We restrict ourselves to
one dimensional models in the discrete case. We consider two type of ergodic
potentials on the half-axis, periodic potentials and random potentials. For
both models, we describe the behavior of the resonances near the real axis for
a large typical sample of the potential. In both cases, the linear density of
their real parts is given by the density of states of the full ergodic system.
While in the periodic case, the resonances distribute on a nice analytic curve
(once their imaginary parts are suitably renormalized), In the random case, the
resonances (again after suitable renormalization of both the real and imaginary
parts) form a two dimensional Poisson cloud
Functionals of the Brownian motion, localization and metric graphs
We review several results related to the problem of a quantum particle in a
random environment.
In an introductory part, we recall how several functionals of the Brownian
motion arise in the study of electronic transport in weakly disordered metals
(weak localization).
Two aspects of the physics of the one-dimensional strong localization are
reviewed : some properties of the scattering by a random potential (time delay
distribution) and a study of the spectrum of a random potential on a bounded
domain (the extreme value statistics of the eigenvalues).
Then we mention several results concerning the diffusion on graphs, and more
generally the spectral properties of the Schr\"odinger operator on graphs. The
interest of spectral determinants as generating functions characterizing the
diffusion on graphs is illustrated.
Finally, we consider a two-dimensional model of a charged particle coupled to
the random magnetic field due to magnetic vortices. We recall the connection
between spectral properties of this model and winding functionals of the planar
Brownian motion.Comment: Review article. 50 pages, 21 eps figures. Version 2: section 5.5 and
conclusion added. Several references adde
Solitary Wave-Series Solutions to Non-Linear Schrodinger Equations
ABSTRACT: In this paper, higher-order dispersive non-linear Schrodinger equations are studied. Their solitary wave-series solutions with continuity of the derivatives and specific discontinuity of the derivatives at the crest are presented. Furthermore, convergence of the series’ solutions is also validated and discussed with the help of graphs.
ABSTRAK: Kertas ini mengkaji persamaan Schrodinger serakan taklinear turutan tinggi. Penyelesaian siri-gelombang tunggalnya dengan kamiran berterusan dan kamiran tak berterusan pada maksimum telah dibentangkan. Penumpuan penyelesaian siri juga telah diperiksa dan dibincangkan dengan bantuan graf-graf.
KEYWORDS: Schrodinger equation; solitary wave-series solution; continuity and discontinuity of derivatives at cres
Resonances for large one-dimensional "ergodic" systems
The present paper is devoted to the study of resonances for one-dimensional
quantum systems with a potential that is the restriction to some large box of
an ergodic potential. For discrete models both on a half-line and on the whole
line, we study the distributions of the resonances in the limit when the size
of the box where the potential does not vanish goes to infinity. For periodic
and random potentials, we analyze how the spectral theory of the limit operator
influences the distribution of the resonances.Comment: Many typos were correcte
Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity
We consider a multichannel wire with a disordered region of length and a
reflecting boundary. The reflection of a wave of frequency is
described by the scattering matrix , encoding the
probability amplitudes to be scattered from one channel to another. The
Wigner-Smith time delay matrix is another important matrix
encoding temporal aspects of the scattering process. In order to study its
statistical properties, we split the scattering matrix in terms of two unitary
matrices,
(with in the presence of TRS), and
introduce a novel symmetrisation procedure for the Wigner-Smith matrix:
,
where is the wave vector and the group velocity. We demonstrate that
can be expressed under the form of an exponential
functional of a matrix Brownian motion. For semi-infinite wires, ,
using a matricial extension of the Dufresne identity, we recover
straightforwardly the joint distribution for 's eigenvalues of
Brouwer and Beenakker [Physica E 9 (2001) p. 463]. For finite length , the
exponential functional representation is used to calculate the first moments
,
and
. Finally we derive a
partial differential equation for the resolvent
in the large limit.Comment: 30 pages, LaTe