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

On the spectrum of a bent chain graph

We study Schr\"odinger operators on an infinite quantum graph of a chain form which consists of identical rings connected at the touching points by $\delta$-couplings with a parameter $\alpha\in\R$. If the graph is "straight", i.e. periodic with respect to ring shifts, its Hamiltonian has a band spectrum with all the gaps open whenever $\alpha\ne 0$. We consider a "bending" deformation of the chain consisting of changing one position at a single ring and show that it gives rise to eigenvalues in the open spectral gaps. We analyze dependence of these eigenvalues on the coupling $\alpha$ and the "bending angle" as well as resonances of the system coming from the bending. We also discuss the behaviour of the eigenvalues and resonances at the edges of the spectral bands.Comment: LaTeX, 23 pages with 7 figures; minor changes, references added; to appear in J. Phys. A: Math. Theo

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

Dispersion for the Schr\"odinger Equation on Networks

In this paper we consider the Schr\"odinger equation on a network formed by a tree with the last generation of edges formed by infinite strips. We give an explicit description of the solution of the linear Schr\"odinger equation with constant coefficients. This allows us to prove dispersive estimates, which in turn are useful for solving the nonlinear Schr\"odinger equation. The proof extends also to the laminar case of positive step-function coefficients having a finite number of discontinuities.Comment: 16 pages, 2 figure

Scattering through a straight quantum waveguide with combined boundary conditions

Scattering through a straight two-dimensional quantum waveguide Rx(0,d) with Dirichlet boundary conditions on (-\infty,0)x{y=0} \cup (0,\infty)x{y=d} and Neumann boundary condition on (-infty,0)x{y=d} \cup (0,\infty)x{y=0} is considered using stationary scattering theory. The existence of a matching conditions solution at x=0 is proved. The use of stationary scattering theory is justified showing its relation to the wave packets motion. As an illustration, the matching conditions are also solved numerically and the transition probabilities are shown.Comment: 26 pages, 3 figure

Sigmund Exner's (1887) einige beobachtungen über bewegungsnachbilder (some observations on movement aftereffects):an illustrated translation with commentary

In his original contribution, Exner’s principal concern was a comparison between the properties of different aftereffects, and particularly to determine whether aftereffects of motion were similar to those of color and whether they could be encompassed within a unified physiological framework. Despite the fact that he was unable to answer his main question, there are some excellent—so far unknown—contributions in Exner’s paper. For example, he describes observations that can be related to binocular interaction, not only in motion aftereffects but also in rivalry. To the best of our knowledge, Exner provides the first description of binocular rivalry induced by differently moving patterns in each eye, for motion as well as for their aftereffects. Moreover, apart from several known, but beautifully addressed, phenomena he makes a clear distinction between motion in depth based on stimulus properties and motion in depth based on the interpretation of motion. That is, the experience of movement, as distinct from the perception of movement. The experience, unlike the perception, did not result in a motion aftereffect in depth

On the spectrum of the Laplace operator of metric graphs attached at a vertex -- Spectral determinant approach

We consider a metric graph $\mathcal{G}$ made of two graphs $\mathcal{G}_1$ and $\mathcal{G}_2$ attached at one point. We derive a formula relating the spectral determinant of the Laplace operator $S_\mathcal{G}(\gamma)=\det(\gamma-\Delta)$ in terms of the spectral determinants of the two subgraphs. The result is generalized to describe the attachment of $n$ graphs. The formulae are also valid for the spectral determinant of the Schr\"odinger operator $\det(\gamma-\Delta+V(x))$.Comment: LaTeX, 8 pages, 7 eps figures, v2: new appendix, v3: discussions and ref adde

A general approximation of quantum graph vertex couplings by scaled Schroedinger operators on thin branched manifolds

We demonstrate that any self-adjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schroedinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local change of graph topology in the vicinity of the vertex; the approximation scheme constructed on the graph is subsequently `lifted' to the manifold. For the corresponding operator a norm-resolvent convergence is proved, with the natural identification map, as the tube diameters tend to zero.Comment: 19 pages, one figure; introduction amended and some references added, to appear in CM

In Vitro Inhibition of Coagulase-Negative Staphylococci by Vancomycin/Aminoglycoside-Loaded Cement Spacers

Background:: Successful treatment of allograft infections by the temporary implantation of an antibiotic-loaded polymethylmethacrylate cement spacer depends on the diffusion of antibiotics out of the cement and inhibition of bacterial growth in the surrounding tissue. We investigated with an in vitro model how long antibiotics are released by the cement and if gentamicin-resistant coagulase-negative staphylococci (CNS) are inhibited by vancomycin mixed with the gentamicin-loaded cement. Materials and Methods:: Four formulations of antibiotic-loaded cement disks, i.e. gentamicin, tobramycin, vancomycin and tobramycin combined with vancomycin, respectively, were used to test the inhibition of eight isolates of Staphylococcus epidermidis and two reference strains of Staphylococcus aureus by an agar diffusion test on Mueller-Hinton (MH) agar similar to the routine laboratory disk diffusion method. Moreover, cement spacer cylinders loaded with gentamicin alone or combined with vancomycin were submerged in MH agar for weeks and the capacity to inhibit five different isolates of S. epidermidis was measured. Results:: The size of the inhibition zones around the antibiotic-loaded cement disks correlated with the minimal inhibitory concentration (MIC) of the antibiotics against the tested strains. All five strains of S. epidermidis were inhibited by vancomycin-loaded cement spacers for at least 30 days. However, two gentamicin-resistant S. epidermidis strains with MICs of 4 mg/l and 16 mg/l could not be inhibited longer than 3 days by the gentamicin-loaded cement spacer. Conclusion:: The in vitro data suggest that antibiotic-loaded cement spacers inhibit susceptible bacteria for 4-6 weeks. The addition of vancomycin to commercial aminoglycoside-loaded cements might be helpful in allograft infections in tumor patients to inhibit a broad range of bacteria including gentamicin-resistant CNS very commonly found in such infection

Avoided crossings in mesoscopic systems: electron propagation on a non-uniform magnetic cylinder

We consider an electron constrained to move on a surface with revolution symmetry in the presence of a constant magnetic field $B$ parallel to the surface axis. Depending on $B$ and the surface geometry the transverse part of the spectrum typically exhibits many crossings which change to avoided crossings if a weak symmetry breaking interaction is introduced. We study the effect of such perturbations on the quantum propagation. This problem admits a natural reformulation to which tools from molecular dynamics can be applied. In turn, this leads to the study of a perturbation theory for the time dependent Born-Oppenheimer approximation