Non-Weyl asymptotics for quantum graphs with general coupling conditions

Inspired by a recent result of Davies and Pushnitski, we study resonance asymptotics of quantum graphs with general coupling conditions at the vertices. We derive a criterion for the asymptotics to be of a non-Weyl character. We show that for balanced vertices with permutation-invariant couplings the asymptotics is non-Weyl only in case of Kirchhoff or anti-Kirchhoff conditions, while for graphs without permutation numerous examples of non-Weyl behaviour can be constructed. Furthermore, we present an insight helping to understand what makes the Kirchhoff/anti-Kirchhoff coupling particular from the resonance point of view. Finally, we demonstrate a generalization to quantum graphs with nonequal edge weights.Comment: minor changes, to appear in Pierre Duclos memorial issue of J. Phys. A: Math. Theo

A lower bound to the spectral threshold in curved tubes

We consider the Laplacian in curved tubes of arbitrary cross-section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube.Comment: LaTeX, 13 pages; to appear in R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sc

A single-mode quantum transport in serial-structure geometric scatterers

We study transport in quantum systems consisting of a finite array of N identical single-channel scatterers. A general expression of the S matrix in terms of the individual-element data obtained recently for potential scattering is rederived in this wider context. It shows in particular how the band spectrum of the infinite periodic system arises in the limit $N\to\infty$. We illustrate the result on two kinds of examples. The first are serial graphs obtained by chaining loops or T-junctions. A detailed discussion is presented for a finite-periodic "comb"; we show how the resonance poles can be computed within the Krein formula approach. Another example concerns geometric scatterers where the individual element consists of a surface with a pair of leads; we show that apart of the resonances coming from the decoupled-surface eigenvalues such scatterers exhibit the high-energy behavior typical for the delta' interaction for the physically interesting couplings.Comment: 36 pages, a LaTeX source file with 2 TeX drawings, 3 ps and 3 jpeg figures attache

Bound states in point-interaction star-graphs

We discuss the discrete spectrum of the Hamiltonian describing a two-dimensional quantum particle interacting with an infinite family of point interactions. We suppose that the latter are arranged into a star-shaped graph with N arms and a fixed spacing between the interaction sites. We prove that the essential spectrum of this system is the same as that of the infinite straight "polymer", but in addition there are isolated eigenvalues unless N=2 and the graph is a straight line. We also show that the system has many strongly bound states if at least one of the angles between the star arms is small enough. Examples of eigenfunctions and eigenvalues are computed numerically.Comment: 17 pages, LaTeX 2e with 9 eps figure

Lieb-Thirring inequalities for geometrically induced bound states

We prove new inequalities of the Lieb-Thirring type on the eigenvalues of Schr\"odinger operators in wave guides with local perturbations. The estimates are optimal in the weak-coupling case. To illustrate their applications, we consider, in particular, a straight strip and a straight circular tube with either mixed boundary conditions or boundary deformations.Comment: LaTeX2e, 14 page

Quantum mechanics of layers with a finite number of point perturbations

We study spectral and scattering properties of a spinless quantum particle confined to an infinite planar layer with hard walls containing a finite number of point perturbations. A solvable character of the model follows from the explicit form of the Hamiltonian resolvent obtained by means of Krein's formula. We prove the existence of bound states, demonstrate their properties, and find the on-shell scattering operator. Furthermore, we analyze the situation when the system is put into a homogeneous magnetic field perpendicular to the layer; in that case the point interactions generate eigenvalues of a finite multiplicity in the gaps of the free Hamiltonian essential spectrum.Comment: LateX 2e, 48 pages, with 3 ps and 3 eps figure

Exponential splitting of bound states in a waveguide with a pair of distant windows

We consider Laplacian in a straight planar strip with Dirichlet boundary which has two Neumann ``windows'' of the same length the centers of which are $2l$ apart, and study the asymptotic behaviour of the discrete spectrum as $l\to\infty$. It is shown that there are pairs of eigenvalues around each isolated eigenvalue of a single-window strip and their distances vanish exponentially in the limit $l\to\infty$. We derive an asymptotic expansion also in the case where a single window gives rise to a threshold resonance which the presence of the other window turns into a single isolated eigenvalue