An isoperimetric problem for leaky loops and related mean-chord inequalities

We consider a class of Hamiltonians in $L^2(\R^2)$ with attractive interaction supported by piecewise $C^2$ smooth loops $\Gamma$ of a fixed length $L$, formally given by $-\Delta-\alpha\delta(x-\Gamma)$ with $\alpha>0$. It is shown that the ground state of this operator is locally maximized by a circular $\Gamma$. We also conjecture that this property holds globally and show that the problem is related to an interesting family of geometric inequalities concerning mean values of chords of $\Gamma$.Comment: LaTeX, 16 page

An isoperimetric problem for point interactions

We consider Hamiltonian with $N$ point interactions in $\R^d, d=2,3,$ all with the same coupling constant, placed at vertices of an equilateral polygon \PP_N. It is shown that the ground state energy is locally maximized by a regular polygon. The question whether the maximum is global is reduced to an interesting geometric problem.Comment: LaTeX 2e, 10 page

Leaky quantum graphs: approximations by point interaction Hamiltonians

We prove an approximation result showing how operators of the type $-\Delta -\gamma \delta (x-\Gamma)$ in $L^2(\mathbb{R}^2)$, where $\Gamma$ is a graph, can be modeled in the strong resolvent sense by point-interaction Hamiltonians with an appropriate arrangement of the $\delta$ potentials. The result is illustrated on finding the spectral properties in cases when $\Gamma$ is a ring or a star. Furthermore, we use this method to indicate that scattering on an infinite curve $\Gamma$ which is locally close to a loop shape or has multiple bends may exhibit resonances due to quantum tunneling or repeated reflections.Comment: LaTeX 2e, 31 pages with 18 postscript figure

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

On the number of particles which a curved quantum waveguide can bind

We discuss the discrete spectrum of N particles in a curved planar waveguide. If they are neutral fermions, the maximum number of particles which the waveguide can bind is given by a one-particle Birman-Schwinger bound in combination with the Pauli principle. On the other hand, if they are charged, e.g., electrons in a bent quantum wire, the Coulomb repulsion plays a crucial role. We prove a sufficient condition under which the discrete spectrum of such a system is empty.Comment: a LateX file, 12 page

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

Tibiotalare Arthrodese bei angeborener Fibulaaplasie

Zusammenfassung: Das Problem: 10-jähriger Patient mit angeborenen Fibulaaplasien beidseits. Auf der linken Seite extreme Valgusfehlstellung des dreistrahligen Fußes im oberen Sprunggelenk mit schmerzhaften Funktionsstörungen beim Stehen und Gehen, die mit orthopädietechnischen Maßnahmen nicht mehr befriedigend beseitigt werden konnten und zu verschiedenen therapeutischen Überlegungen, wie Amputation des Fußes, supramalleoläre Umstellungsosteotomie und tibiotalare Arthrodese, Anlass gaben. Die Lösung: Tibiotalare korrigierende Arthrodese mit einem sog. Minifixateur unter Erhaltung der distalen Epiphysenscheibe der Tibia. Operationstechnik: Erste Inzision auf der Medialseite zur Darstellung der Beugesehnen und des Gefäß-Nerven-Bündels unter Schonung des Nervus suralis und der Vena saphena parva. Freilegung des Innenknöchels nach Durchtrennung seiner Band- und Kapselverbindungen sowie Lokalisation des oberen Sprunggelenkspalts. Längsschnitt auf der Lateralseite des oberen Sprunggelenks. Z-förmige Verlängerung der einzig angelegten Peronealsehne. Eröffnung des oberen Sprunggelenks auf der Lateral- und Ventralseite. Resektion der Gelenkflächen des Talus und der distalen Tibia entsprechend einer Operationsskizze, nach der eine achsengerechte Unterstellung des Rückfußes unter die Tibialängsachse in Rechtwinkelstellung erreicht wird. Einbringen eines Kirschner-Drahts von der Fußsohle in die Tibia zur temporären Fixation der erreichten Korrektur. Anlegen des sog. Minifixateurs: Ein gewindetragender Kirschner- Draht wird durch die Synostose, ein zweiter durch die Epiphyse und ein dritter durch das proximale Tibiadrittel gebohrt. Nach Montage des Fixateurrahmens Kompression der Resektionsflächen und Distraktion zwischen dem proximalen und mittleren Kirschner-Draht. Ergebnis: Im Alter von 16 Jahren trägt der Patient einen Innenschuh in normalen Konfektionsschuhen; er ist schmerzfrei und nimmt an allen Aktivitäten des Alltags teil. Das Längenwachstum der Tibia ist nicht beeinträchtigt worde

Multiple bound states in scissor-shaped waveguides

We study bound states of the two-dimensional Helmholtz equations with Dirichlet boundary conditions in an open geometry given by two straight leads of the same width which cross at an angle $\theta$. Such a four-terminal junction with a tunable $\theta$ can realized experimentally if a right-angle structure is filled by a ferrite. It is known that for $\theta=90^o$ there is one proper bound state and one eigenvalue embedded in the continuum. We show that the number of eigenvalues becomes larger with increasing asymmetry and the bound-state energies are increasing as functions of $\theta$ in the interval $(0,90^o)$. Moreover, states which are sufficiently strongly bent exist in pairs with a small energy difference and opposite parities. Finally, we discuss how with increasing $\theta$ the bound states transform into the quasi-bound states with a complex wave vector.Comment: 6 pages, 6 figure

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