4,481 research outputs found
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
The quantized Hall effect in the presence of resistance fluctuations
We present an experimental study of mesoscopic, two-dimensional electronic
systems at high magnetic fields. Our samples, prepared from a low-mobility
InGaAs/InAlAs wafer, exhibit reproducible, sample specific, resistance
fluctuations. Focusing on the lowest Landau level we find that, while the
diagonal resistivity displays strong fluctuations, the Hall resistivity is free
of fluctuations and remains quantized at its value, . This is
true also in the insulating phase that terminates the quantum Hall series.
These results extend the validity of the semicircle law of conductivity in the
quantum Hall effect to the mesoscopic regime.Comment: Includes more data, changed discussio
Linear response formula for piecewise expanding unimodal maps
The average R(t) of a smooth function with respect to the SRB measure of a
smooth one-parameter family f_t of piecewise expanding interval maps is not
always Lipschitz. We prove that if f_t is tangent to the topological class of
f_0, then R(t) is differentiable at zero, and the derivative coincides with the
resummation previously proposed by the first named author of the (a priori
divergent) series given by Ruelle's conjecture.Comment: We added Theorem 7.1 which shows that the horizontality condition is
necessary. The paper "Smooth deformations..." containing Thm 2.8 is now
available on the arxiv; see also Corrigendum arXiv:1205.5468 (to appear
Nonlinearity 2012
Diamagnetism of quantum gases with singular potentials
We consider a gas of quasi-free quantum particles confined to a finite box,
subjected to singular magnetic and electric fields. We prove in great
generality that the finite volume grand-canonical pressure is jointly analytic
in the chemical potential ant the intensity of the external magnetic field. We
also discuss the thermodynamic limit
Breit Equation with Form Factors in the Hydrogen Atom
The Breit equation with two electromagnetic form-factors is studied to obtain
a potential with finite size corrections. This potential with proton structure
effects includes apart from the standard Coulomb term, the Darwin term,
retarded potentials, spin-spin and spin-orbit interactions corresponding to the
fine and hyperfine structures in hydrogen atom. Analytical expressions for the
hyperfine potential with form factors and the subsequent energy levels
including the proton structure corrections are given using the dipole form of
the form factors. Numerical results are presented for the finite size
corrections in the 1S and 2S hyperfine splittings in the hydrogen atom, the
Sternheim observable and the 2S and 2P hyperfine splittings in muonic
hydrogen. Finally, a comparison with some other existing methods in literature
is presented.Comment: 24 pages, Latex, extended version, title change
Dipole binding in a cosmic string background due to quantum anomalies
We propose quantum dynamics for the dipole moving in cosmic string background
and show that the classical scale symmetry of a particle moving in cosmic
string background is still restored even in the presence of dipole moment of
the particle. However, we show that the classical scale symmetry is broken due
to inequivalent quantization of the the non-relativistic system. The
consequence of this quantum anomaly is the formation of bound state in the
interval \xi\in(-1,1). The inequivalent quantization is characterized by a
1-parameter family of self-adjoint extension parameter \Sigma. We show that
within the interval \xi\in(-1,1), cosmic string with zero radius can bind the
dipole and the dipole does not fall into the singularity.Comment: Accepted for publication in Phys. Rev.
Rigorous investigation of the reduced density matrix for the ideal Bose gas in harmonic traps by a loop-gas-like approach
In this paper, we rigorously investigate the reduced density matrix (RDM)
associated to the ideal Bose gas in harmonic traps. We present a method based
on a sum-decomposition of the RDM allowing to treat not only the isotropic
trap, but also general anisotropic traps. When focusing on the isotropic trap,
the method is analogous to the loop-gas approach developed by W.J. Mullin in
[38]. Turning to the case of anisotropic traps, we examine the RDM for some
anisotropic trap models corresponding to some quasi-1D and quasi-2D regimes.
For such models, we bring out an additional contribution in the local density
of particles which arises from the mesoscopic loops. The close connection with
the occurrence of generalized-BEC is discussed. Our loop-gas-like approach
provides relevant information which can help guide numerical investigations on
highly anisotropic systems based on the Path Integral Monte Carlo (PIMC)
method.Comment: v3: Minor modifications of v2. v2: Major modifications: the former
version (v1) has been completely rewritten. New results concerning the
anisotropic traps and generalized Bose-Einstein condensation have been added.
The connection with the loop-gas approach is further discussed. 40 page
Linear vs. nonlinear effects for nonlinear Schrodinger equations with potential
We review some recent results on nonlinear Schrodinger equations with
potential, with emphasis on the case where the potential is a second order
polynomial, for which the interaction between the linear dynamics caused by the
potential, and the nonlinear effects, can be described quite precisely. This
includes semi-classical regimes, as well as finite time blow-up and scattering
issues. We present the tools used for these problems, as well as their
limitations, and outline the arguments of the proofs.Comment: 20 pages; survey of previous result
The energy of the analytic lump solution in SFT
In a previous paper a method was proposed to find exact analytic solutions of
open string field theory describing lower dimensional lumps, by incorporating
in string field theory an exact renormalization group flow generated by a
relevant operator in a worldsheet CFT. In this paper we compute the energy of
one such solution, which is expected to represent a D24 brane. We show, both
numerically and analytically, that its value corresponds to the theoretically
expected one.Comment: 45 pages, former section 2 suppressed, Appendix D added, comments and
references added, typos corrected. Erratum adde
New summing algorithm using ensemble computing
We propose an ensemble algorithm, which provides a new approach for
evaluating and summing up a set of function samples. The proposed algorithm is
not a quantum algorithm, insofar it does not involve quantum entanglement. The
query complexity of the algorithm depends only on the scaling of the
measurement sensitivity with the number of distinct spin sub-ensembles. From a
practical point of view, the proposed algorithm may result in an exponential
speedup, compared to known quantum and classical summing algorithms. However in
general, this advantage exists only if the total number of function samples is
below a threshold value which depends on the measurement sensitivity.Comment: 13 pages, 0 figures, VIth International Conference on Quantum
Communication, Measurement and Computing (Boston, 2002
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