Graph-like asymptotics for the Dirichlet Laplacian in connected tubular domains

We consider the Dirichlet Laplacian in a waveguide of uniform width and infinite length which is ideally divided into three parts: a "vertex region", compactly supported and with non zero curvature, and two "edge regions" which are semi-infinite straight strips. We make the waveguide collapse onto a graph by squeezing the edge regions to half-lines and the vertex region to a point. In a setting in which the ratio between the width of the waveguide and the longitudinal extension of the vertex region goes to zero, we prove the convergence of the operator to a selfadjoint realization of the Laplacian on a two edged graph. In the limit operator, the boundary conditions in the vertex depend on the spectral properties of an effective one dimensional Hamiltonian associated to the vertex region.Comment: Major revision. Reviewed introduction. Changes in Th. 1, Th. 2, and Th. 3. Updated references. 23 page

Nontrivial edge coupling from a Dirichlet network squeezing: the case of a bent waveguide

In distinction to the Neumann case the squeezing limit of a Dirichlet network leads in the threshold region generically to a quantum graph with disconnected edges, exceptions may come from threshold resonances. Our main point in this paper is to show that modifying locally the geometry we can achieve in the limit a nontrivial coupling between the edges including, in particular, the class of $\delta$-type boundary conditions. We work out an illustration of this claim in the simplest case when a bent waveguide is squeezed.Comment: LaTeX, 16 page

Relative partition function of Coulomb plus delta interaction

The relative partition function and the relative zeta function of the perturbation of the Laplace operator by a Coulomb potential plus a point interaction centered in the origin is discussed. Applications to the study of the Casimir effect are indicated.Comment: Minor misprints corrected. 24 page

Quasi-1D Bose-Einstein condensates in the dimensional crossover regime

We study theoretically the dimensional crossover from a three-dimensional elongated condensate to a one-dimensional condensate as the transverse degrees of freedom get frozen by tight confinement, in the limit of small density fluctuations, i.e. for a strongly degenerate gas. We compute analytically the radially integrated density profile at low temperatures using a local density approximation, and study the behavior of phase fluctuations with the transverse confinement. Previous studies of phase fluctuations in trapped gases have either focused on the 3D elongated regimes or on the 1D regime. The present approach recovers these previous results and is able to interpolate between them. We show in particular that in this strongly degenerate limit the shape of the spatial correlation function is insensitive to the transverse regime of confinement, pointing out to an almost universal behavior of phase fluctuations in elongated traps

Perturbations of eigenvalues embedded at threshold: one, two and three dimensional solvable models

We examine perturbations of eigenvalues and resonances for a class of multi-channel quantum mechanical model-Hamiltonians describing a particle interacting with a localized spin in dimension $d=1,2,3$. We consider unperturbed Hamiltonians showing eigenvalues and resonances at the threshold of the continuous spectrum and we analyze the effect of various type of perturbations on the spectral singularities. We provide algorithms to obtain convergent series expansions for the coordinates of the singularities.Comment: 20 page

Time dependent delta-prime interactions in dimension one

We solve the Cauchy problem for the Schr\"odinger equation corresponding to the family of Hamiltonians $H_{\gamma(t)}$ in $L^{2}(\mathbb{R})$ which describes a $\delta'$-interaction with time-dependent strength $1/\gamma(t)$. We prove that the strong solution of such a Cauchy problem exits whenever the map $t\mapsto\gamma(t)$ belongs to the fractional Sobolev space $H^{3/4}(\mathbb{R})$, thus weakening the hypotheses which would be required by the known general abstract results. The solution is expressed in terms of the free evolution and the solution of a Volterra integral equation.Comment: minor changes, 10 page

Effective equation for a system of mechanical oscillators in an acoustic field

We consider a one dimensional evolution problem modeling the dynamics of an acoustic field coupled with a set of mechanical oscillators. We analyze solutions of the system of ordinary and partial differential equations with time-dependent boundary conditions describing the evolution in the limit of a continuous distribution of oscillators.Comment: Improved Theorem 2. Updated introduction and references. Added 1 figure. 11 page

Bounds for the Stieltjes Transform and the Density of States of Wigner Matrices

We consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from [22,10], by removing the logarithmic corrections. As applications, we establish the convergence of the eigenvalue counting functions with the rate $(\log N)/N$ and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bounds improve the results of [22,10,23].Comment: New title, former title "Optimal Bounds on the Stieltjes Transform of Wigner Matrices". Updated reference