Violation of the entropic area law for Fermions

We investigate the scaling of the entanglement entropy in an infinite translational invariant Fermionic system of any spatial dimension. The states under consideration are ground states and excitations of tight-binding Hamiltonians with arbitrary interactions. We show that the entropy of a finite region typically scales with the area of the surface times a logarithmic correction. Thus, in contrast to analogous Bosonic systems, the entropic area law is violated for Fermions. The relation between the entanglement entropy and the structure of the Fermi surface is discussed, and it is proven, that the presented scaling law holds whenever the Fermi surface is finite. This is in particular true for all ground states of Hamiltonians with finite range interactions.Comment: 5 pages, 1 figur

Small-Energy Analysis for the Selfadjoint Matrix Schroedinger Operator on the Half Line

The matrix Schroedinger equation with a selfadjoint matrix potential is considered on the half line with the most general selfadjoint boundary condition at the origin. When the matrix potential is integrable and has a first moment, it is shown that the corresponding scattering matrix is continuous at zero energy. An explicit formula is provided for the scattering matrix at zero energy. The small-energy asymptotics are established also for the corresponding Jost matrix, its inverse, and various other quantities relevant to the corresponding direct and inverse scattering problems.Comment: This published version has been edited to improve the presentation of the result

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

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

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 $\alpha \in \R$ 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

Ferromagnetic Ordering of Energy Levels for Uq(sl2)U_q(\mathfrak{sl}_2) Symmetric Spin Chains

We consider the class of quantum spin chains with arbitrary $U_q(\mathfrak{sl}_2)$-invariant nearest neighbor interactions, sometimes called $\textrm{SU}_q(2)$ for the quantum deformation of $\textrm{SU}(2)$, for $q>0$. 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 $U_q(\mathfrak{sl}_2)$ intertwiners we define. We also state applications to interacting particle processes.Comment: 23 page

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

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 closer look at the uncertainty relation of position and momentum

We consider particles prepared by the von Neumann-L\"uders projection. For those particles the standard deviation of the momentum is discussed. We show that infinite standard deviations are not exceptions but rather typical. A necessary and sufficient condition for finite standard deviations is given. Finally, a new uncertainty relation is derived and it is shown that the latter cannot be improved.Comment: 3 pages, introduction shortened, content unchange