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

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

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

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

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

In search of suitable in vitro models to study germ cell movement across the blood-testis barrier

The movement of preleptotene/leptotene spermatocytes across the blood-testis barrier, also known as the Sertoli cell barrier, during stages VIII to XI of the seminiferous epithelial cycle is one of the most important cellular events taking place in the mammalian testis. Without the passage of spermatocytes, spermatogenesis would be halted, resulting in transient or permanent sterility. Unfortunately, we have very little knowledge on how preleptotene/leptotene spermatocytes cross the blood-testis barrier. While we know cytokines, proteases and androgens to mediate Sertoli cell junction restructuring, most data continue to be derived from experiments using Sertoli cells cultured alone in two dimensions. Thus, additional in vitro models which include germ cells must come into use. In this Commentary, we hope to shed new light on how we may better study spermatocyte movement across the BTB

Cranial and intra-axial metastasis originating from a primary ovarian Dysgerminoma.

Dysgerminomas are aggressive germ cell tumors that typically have a favorable prognosis, especially in patients diagnosed with early stage disease. We recount the history of a 23-year-old woman who was treated for a stage IA ovarian dysgerminoma in November 2017. Postoperatively, the patient was noncompliant insofar as obtaining routine lab evaluations; ten months later, she was diagnosed with a cranial metastasis that extended into the meninges. The patient subsequently underwent a posterior fossa craniotomy and adjuvant etoposide, bleomycin and cisplatin chemotherapy to which she initially responded; however, during cycle 4, she developed pancytopenia whereupon the chemotherapy was summarily discontinued. Thereafter, the patient was surveilled and currently, she remains in clinical remission. Early stage ovarian dysgerminoma, albeit rarely, has the capacity to metastasize to the cranium or brain, further underscoring the significance of employing active follow-up with these patients