2,067 research outputs found
An isoperimetric problem for point interactions
We consider Hamiltonian with point interactions in 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
An isoperimetric problem for leaky loops and related mean-chord inequalities
We consider a class of Hamiltonians in with attractive
interaction supported by piecewise smooth loops of a fixed
length , formally given by with .
It is shown that the ground state of this operator is locally maximized by a
circular . 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 .Comment: LaTeX, 16 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
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
Tibiotalare Arthrodese bei angeborener Fibulaaplasie
Zusammenfassung: Das Problem: 10-jähriger Patient mit angeborenen Fibulaaplasien beidseits. Auf der linken Seite extreme Valgusfehlstellung des dreistrahligen FuĂes im oberen Sprunggelenk mit schmerzhaften FunktionsstĂśrungen beim Stehen und Gehen, die mit orthopädietechnischen MaĂnahmen nicht mehr befriedigend beseitigt werden konnten und zu verschiedenen therapeutischen Ăberlegungen, wie Amputation des FuĂes, supramalleoläre Umstellungsosteotomie und tibiotalare Arthrodese, Anlass gaben. Die LĂśsung: 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 GefäĂ-Nerven-BĂźndels unter Schonung des Nervus suralis und der Vena saphena parva. Freilegung des InnenknĂśchels nach Durchtrennung seiner Band- und Kapselverbindungen sowie Lokalisation des oberen Sprunggelenkspalts. Längsschnitt auf der Lateralseite des oberen Sprunggelenks. Z-fĂśrmige Verlängerung der einzig angelegten Peronealsehne. ErĂśffnung 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 RĂźckfuĂes unter die Tibialängsachse in Rechtwinkelstellung erreicht wird. Einbringen eines Kirschner-Drahts von der FuĂsohle 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
Leaky quantum graphs: approximations by point interaction Hamiltonians
We prove an approximation result showing how operators of the type in , where is a graph,
can be modeled in the strong resolvent sense by point-interaction Hamiltonians
with an appropriate arrangement of the potentials. The result is
illustrated on finding the spectral properties in cases when is a ring
or a star. Furthermore, we use this method to indicate that scattering on an
infinite curve 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 . 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
SchrĂśdinger operators with δ and δâ˛-potentials supported on hypersurfaces
Self-adjoint SchrĂśdinger operators with δ and δâ˛-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the BirmanâSchwinger principle and a variant of Kreinâs formula are shown. Furthermore, Schattenâvon Neumann type estimates for the differences of the powers of the resolvents of the SchrĂśdinger operators with δ and δâ˛-potentials, and the SchrĂśdinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed SchrĂśdinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity
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
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
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