729 research outputs found
Continued fraction solution of Krein's inverse problem
The spectral data of a vibrating string are encoded in its so-called
characteristic function. We consider the problem of recovering the distribution
of mass along the string from its characteristic function. It is well-known
that Stieltjes' continued fraction provides a solution of this inverse problem
in the particular case where the distribution of mass is purely discrete. We
show how to adapt Stieltjes' method to solve the inverse problem for a related
class of strings. An application to the excursion theory of diffusion processes
is presented.Comment: 18 pages, 2 figure
Maximal violation of Bell inequalities by position measurements
We show that it is possible to find maximal violations of the CHSH-Bell
inequality using only position measurements on a pair of entangled
non-relativistic free particles. The device settings required in the CHSH
inequality are done by choosing one of two times at which position is measured.
For different assignments of the "+" outcome to positions, namely to an
interval, to a half line, or to a periodic set, we determine violations of the
inequalities, and states where they are attained. These results have
consequences for the hidden variable theories of Bohm and Nelson, in which the
two-time correlations between distant particle trajectories have a joint
distribution, and hence cannot violate any Bell inequality.Comment: 13 pages, 4 figure
Optimization of quasi-normal eigenvalues for Krein-Nudelman strings
The paper is devoted to optimization of resonances for Krein strings with
total mass and statical moment constraints. The problem is to design for a
given a string that has a resonance on the line \alpha + \i
\R with a minimal possible modulus of the imaginary part. We find optimal
resonances and strings explicitly.Comment: 9 pages, these results were extracted in a slightly modified form
from the earlier e-print arXiv:1103.4117 [math.SP] following an advise of a
journal's refere
Gauss Sums and Quantum Mechanics
By adapting Feynman's sum over paths method to a quantum mechanical system
whose phase space is a torus, a new proof of the Landsberg-Schaar identity for
quadratic Gauss sums is given. In contrast to existing non-elementary proofs,
which use infinite sums and a limiting process or contour integration, only
finite sums are involved. The toroidal nature of the classical phase space
leads to discrete position and momentum, and hence discrete time. The
corresponding `path integrals' are finite sums whose normalisations are derived
and which are shown to intertwine cyclicity and discreteness to give a finite
version of Kelvin's method of images.Comment: 14 pages, LaTe
Asymptotic analysis for the generalized langevin equation
Various qualitative properties of solutions to the generalized Langevin
equation (GLE) in a periodic or a confining potential are studied in this
paper. We consider a class of quasi-Markovian GLEs, similar to the model that
was introduced in \cite{EPR99}. Geometric ergodicity, a homogenization theorem
(invariance principle), short time asymptotics and the white noise limit are
studied. Our proofs are based on a careful analysis of a hypoelliptic operator
which is the generator of an auxiliary Markov process. Systematic use of the
recently developed theory of hypocoercivity \cite{Vil04HPI} is made.Comment: 27 pages, no figures. Submitted to Nonlinearity
A seesaw-lever force-balancing suspension design for space and terrestrial gravity-gradient sensing
We present the design, fabrication, and characterization of a seesaw-lever force-balancing suspension for a silicon gravity-gradient sensor, a gravity gradiometer, that is capable of operation over a range of gravity from 0 to 1 g. This allows for both air and space deployment after ground validation. An overall rationale for designing a microelectromechanical systems(MEMS) gravity gradiometer is developed, indicating that a gravity gradiometer based on a torsion-balance, rather than a differential-accelerometer, provides the best approach. The fundamental micromachined element, a seesaw-lever force-balancing suspension, is designed with a low fundamental frequency for in-plane rotation to response gravity gradient but with good rejection of all cross-axis modes. During operation under 1 g, a gravitational force is axially loaded on two straight-beams that perform as a stiff fulcrum for the mass-connection lever without affecting sensitive in-plane rotational sensing. The dynamics of this suspension are analysed by both closed-form and finite element analysis, with good agreement between the two. The suspension has been fabricated using through-wafer deep reactive-ion etching and the dynamics verified both in air and vacuum. The sensitivity of a gravity gradiometer built around this suspension will be dominated by thermal noise, contributing in this case a noise floor of around 10 E/Hz−−−√10 E/Hz (1 E = 10−9/s2) in vacuum. Compared with previous conventional gravity gradiometers, this suspension allows a gradiometer of performance within an order of magnitude but greatly reduced volume and weight. Compared with previous MEMS gravity gradiometers, our design has the advantage of functionality under Earth gravity
A local-global principle for linear dependence of noncommutative polynomials
A set of polynomials in noncommuting variables is called locally linearly
dependent if their evaluations at tuples of matrices are always linearly
dependent. By a theorem of Camino, Helton, Skelton and Ye, a finite locally
linearly dependent set of polynomials is linearly dependent. In this short note
an alternative proof based on the theory of polynomial identities is given. The
method of the proof yields generalizations to directional local linear
dependence and evaluations in general algebras over fields of arbitrary
characteristic. A main feature of the proof is that it makes it possible to
deduce bounds on the size of the matrices where the (directional) local linear
dependence needs to be tested in order to establish linear dependence.Comment: 8 page
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
Ferromagnetic Ordering of Energy Levels for Symmetric Spin Chains
We consider the class of quantum spin chains with arbitrary
-invariant nearest neighbor interactions, sometimes
called for the quantum deformation of , for
. We derive sufficient conditions for the Hamiltonian to satisfy the
property we call {\em Ferromagnetic Ordering of Energy Levels}. This is the
property that the ground state energy restricted to a fixed total spin subspace
is a decreasing function of the total spin. Using the Perron-Frobenius theorem,
we show sufficient conditions are positivity of all interactions in the dual
canonical basis of Lusztig. We characterize the cone of positive interactions,
showing that it is a simplicial cone consisting of all non-positive linear
combinations of "cascade operators," a special new basis of
intertwiners we define. We also state applications to
interacting particle processes.Comment: 23 page
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