1,637 research outputs found
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
Scattering by local deformations of a straight leaky wire
We consider a model of a leaky quantum wire with the Hamiltonian in , where is a compact
deformation of a straight line. The existence of wave operators is proven and
the S-matrix is found for the negative part of the spectrum. Moreover, we
conjecture that the scattering at negative energies becomes asymptotically
purely one-dimensional, being determined by the local geometry in the leading
order, if is a smooth curve and .Comment: Latex2e, 15 page
Schroedinger operators with singular interactions: a model of tunneling resonances
We discuss a generalized Schr\"odinger operator in , with an attractive singular interaction supported by a
-dimensional hyperplane and a finite family of points. It can be
regarded as a model of a leaky quantum wire and a family of quantum dots if
, or surface waves in presence of a finite number of impurities if .
We analyze the discrete spectrum, and furthermore, we show that the resonance
problem in this setting can be explicitly solved; by Birman-Schwinger method it
is cast into a form similar to the Friedrichs model.Comment: LaTeX2e, 34 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
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
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 . Such a four-terminal
junction with a tunable can realized experimentally if a right-angle
structure is filled by a ferrite. It is known that for 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 in the interval
. Moreover, states which are sufficiently strongly bent exist in
pairs with a small energy difference and opposite parities. Finally, we discuss
how with increasing the bound states transform into the quasi-bound
states with a complex wave vector.Comment: 6 pages, 6 figure
Spectra of soft ring graphs
We discuss of a ring-shaped soft quantum wire modeled by interaction
supported by the ring of a generally nonconstant coupling strength. We derive
condition which determines the discrete spectrum of such systems, and analyze
the dependence of eigenvalues and eigenfunctions on the coupling and ring
geometry. In particular, we illustrate that a random component in the coupling
leads to a localization. The discrete spectrum is investigated also in the
situation when the ring is placed into a homogeneous magnetic field or threaded
by an Aharonov-Bohm flux and the system exhibits persistent currents.Comment: LaTeX 2e, 17 pages, with 10 ps figure
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
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