233 research outputs found
Extended Edge States in Finite Hall Systems
We study edge states of a random Schroedinger operator for an electron
submitted to a magnetic field in a finite macroscopic two dimensional system of
linear dimensions equal to L. The y direction is L-periodic and in the x
direction the electron is confined by two smoothly increasing parallel boundary
potentials. We prove that, with large probability, for an energy range in the
first spectral gap of the bulk Hamiltonian, the spectrum of the full
Hamiltonian consists only on two sets of eigenenergies whose eigenfuntions have
average velocities which are strictly positive/negative, uniformly with respect
to the size of the system. Our result gives a well defined meaning to the
notion of edge states for a finite cylinder with two boundaries, and extends
previous studies on systems with only one boundary.Comment: 24 pages, 1 figure; Submitte
The strong converse theorem for the product-state capacity of quantum channels with ergodic Markovian memory
Establishing the strong converse theorem for a communication channel confirms
that the capacity of that channel, that is, the maximum achievable rate of
reliable information communication, is the ultimate limit of communication over
that channel. Indeed, the strong converse theorem for a channel states that
coding at a rate above the capacity of the channel results in the convergence
of the error to its maximum value 1 and that there is no trade-off between
communication rate and decoding error. Here we prove that the strong converse
theorem holds for the product-state capacity of quantum channels with ergodic
Markovian correlated memory.Comment: 11 pages, single colum
Asymptotic Feynman-Kac formulae for large symmetrised systems of random walks
We study large deviations principles for random processes on the
lattice with finite time horizon under a symmetrised
measure where all initial and terminal points are uniformly given by a random
permutation. That is, given a permutation of elements and a
vector of initial points we let the random processes
terminate in the points and then sum over
all possible permutations and initial points, weighted with an initial
distribution. There is a two-level random mechanism and we prove two-level
large deviations principles for the mean of empirical path measures, for the
mean of paths and for the mean of occupation local times under this symmetrised
measure. The symmetrised measure cannot be written as any product of single
random process distributions. We show a couple of important applications of
these results in quantum statistical mechanics using the Feynman-Kac formulae
representing traces of certain trace class operators. In particular we prove a
non-commutative Varadhan Lemma for quantum spin systems with Bose-Einstein
statistics and mean field interactions.
A special case of our large deviations principle for the mean of occupation
local times of simple random walks has the Donsker-Varadhan rate function
as the rate function for the limit but for finite time . We give an interpretation in quantum statistical mechanics for this
surprising result
Discrete-Time Path Distributions on Hilbert Space
We construct a path distribution representing the kinetic part of the Feynman
path integral at discrete times similar to that defined by Thomas [1], but on a
Hilbert space of paths rather than a nuclear sequence space. We also consider
different boundary conditions and show that the discrete-time Feynman path
integral is well-defined for suitably smooth potentials
Three-dimensional Quantum Slit Diffraction and Diffraction in Time
We study the quantum slit diffraction problem in three dimensions. In the
treatment of diffraction of particles by a slit, it is usually assumed that the
motion perpendicular to the slit is classical. Here we take into account the
effect of the quantum nature of the motion perpendicular to the slit using the
Green function approach [18]. We treat the diffraction of a Gaussian wave
packet for general boundary conditions on the shutter. The difference between
the standard and our three-dimensional slit diffraction models is analogous to
the diffraction in time phenomenon introduced in [16]. We derive corrections to
the standard formula for the diffraction pattern, and we point out situations
in which this might be observable. In particular, we discuss the diffraction in
space and time in the presence of gravity
Three Order Parameters in Quantum XZ Spin-Oscillator Models with Gibbsian Ground States
Quantum models on the hyper-cubic d-dimensional lattice of spin-1/2 particles
interacting with linear oscillators are shown to have three ferromagnetic
ground state order parameters. Two order parameters coincide with the
magnetization in the first and third directions and the third one is a
magnetization in a continuous oscillator variable. The proofs use a generalized
Peierls argument and two Griffiths inequalities. The class of spin-oscillator
Hamiltonians considered manifest maximal ordering in their ground states. The
models have relevance for hydrogen-bond ferroelectrics. The simplest of these
is proven to have a unique Gibbsian ground state.Comment: This is a contribution to the Proc. of the Seventh International
Conference ''Symmetry in Nonlinear Mathematical Physics'', published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA/ v2: sections 1 and 2 have been rewritten,
the main result and the proof have not been change
Characterization of the Spectrum of the Landau Hamiltonian with Delta Impurities
We consider a random Schro\"dinger operator in an external magnetic field.
The random potential consists of delta functions of random strengths situated
on the sites of a regular two-dimensional lattice. We characterize the spectrum
in the lowest N Landau bands of this random Hamiltonian when the magnetic field
is sufficiently strong, depending on N. We show that the spectrum in these
bands is entirely pure point, that the energies coinciding with the Landau
levels are infinitely degenerate and that the eigenfunctions corresponding to
energies in the remainder of the spectrum are localized with a uniformly
bounded localization length. By relating the Hamiltonian to a lattice operator
we are able to use the Aizenman-Molchanov method to prove localization.Comment: To appear in Commun. Math. Phys. (1999
Correlation of clusters: Partially truncated correlation functions and their decay
In this article, we investigate partially truncated correlation functions
(PTCF) of infinite continuous systems of classical point particles with pair
interaction. We derive Kirkwood-Salsburg-type equations for the PTCF and write
the solutions of these equations as a sum of contributions labelled by certain
forests graphs, the connected components of which are tree graphs. We
generalize the method introduced by R.A. Minlos and S.K. Poghosyan (1977) in
the case of truncated correlations. These solutions make it possible to derive
strong cluster properties for PTCF which were obtained earlier for lattice spin
systems.Comment: 31 pages, 2 figures. 2nd revision. Misprints corrected and 1 figure
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