2,673 research outputs found
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
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
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
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
Geometric coupling thresholds in a two-dimensional strip
We consider the Laplacian in a strip with the
boundary condition which is Dirichlet except at the segment of a length of
one of the boundaries where it is switched to Neumann. This operator is known
to have a non-empty and simple discrete spectrum for any . There is a
sequence of critical values at which new eigenvalues emerge
from the continuum when the Neumann window expands. We find the asymptotic
behavior of these eigenvalues around the thresholds showing that the gap is in
the leading order proportional to with an explicit coefficient
expressed in terms of the corresponding threshold-energy resonance
eigenfunction
Schrödinger operators with δ and δ′-potentials supported on hypersurfaces
Self-adjoint Schrödinger operators with δ and δ′-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 Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger 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 Schrödinger 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
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
-couplings with a parameter . 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 . 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 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
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