343 research outputs found

### Resonances from perturbations of quantum graphs with rationally related edges

We discuss quantum graphs consisting of a compact part and semiinfinite
leads. Such a system may have embedded eigenvalues if some edge lengths in the
compact part are rationally related. If such a relation is perturbed these
eigenvalues may turn into resonances; we analyze this effect both generally and
in simple examples.Comment: LaTeX source file with 10 pdf figures, 24 pages; a replaced version
with minor improvements, to appear in J. Phys. A: Math. Theo

### Scattering solutions in a network of thin fibers: small diameter asymptotics

Small diameter asymptotics is obtained for scattering solutions in a network
of thin fibers. The asymptotics is expressed in terms of solutions of related
problems on the limiting quantum graph. We calculate the Lagrangian gluing
conditions at vertices for the problems on the limiting graph. If the frequency
of the incident wave is above the bottom of the absolutely continuous spectrum,
the gluing conditions are formulated in terms of the scattering data for each
individual junction of the network

### On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators

The article is devoted to the following question. Consider a periodic
self-adjoint difference (differential) operator on a graph (quantum graph) G
with a co-compact free action of the integer lattice Z^n. It is known that a
local perturbation of the operator might embed an eigenvalue into the
continuous spectrum (a feature uncommon for periodic elliptic operators of
second order). In all known constructions of such examples, the corresponding
eigenfunction is compactly supported. One wonders whether this must always be
the case. The paper answers this question affirmatively. What is more
surprising, one can estimate that the eigenmode must be localized not far away
from the perturbation (in a neighborhood of the perturbation's support, the
width of the neighborhood determined by the unperturbed operator only).
The validity of this result requires the condition of irreducibility of the
Fermi (Floquet) surface of the periodic operator, which is expected to be
satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio

### Quantum Graphs II: Some spectral properties of quantum and combinatorial graphs

The paper deals with some spectral properties of (mostly infinite) quantum
and combinatorial graphs. Quantum graphs have been intensively studied lately
due to their numerous applications to mesoscopic physics, nanotechnology,
optics, and other areas.
A Schnol type theorem is proven that allows one to detect that a point
belongs to the spectrum when a generalized eigenfunction with an subexponential
growth integral estimate is available. A theorem on spectral gap opening for
``decorated'' quantum graphs is established (its analog is known for the
combinatorial case). It is also shown that if a periodic combinatorial or
quantum graph has a point spectrum, it is generated by compactly supported
eigenfunctions (``scars'').Comment: 4 eps figures, LATEX file, 21 pages Revised form: a cut-and-paste
blooper fixe

### Quasiperiodic surface Maryland models on quantum graphs

We study quantum graphs corresponding to isotropic lattices with
quasiperiodic coupling constants given by the same expressions as the
coefficients of the discrete surface Maryland model. The absolutely continuous
and the pure point spectra are described. It is shown that the transition
between them is governed by the Hill operator corresponding to the edge
potential.Comment: 12 page

### Range descriptions for the spherical mean Radon transform

The transform considered in the paper averages a function supported in a ball
in \RR^n over all spheres centered at the boundary of the ball. This Radon
type transform arises in several contemporary applications, e.g. in
thermoacoustic tomography and sonar and radar imaging. Range descriptions for
such transforms are important in all these areas, for instance when dealing
with incomplete data, error correction, and other issues. Four different types
of complete range descriptions are provided, some of which also suggest
inversion procedures. Necessity of three of these (appropriately formulated)
conditions holds also in general domains, while the complete discussion of the
case of general domains would require another publication.Comment: LATEX file, 55 pages, two EPS figure

### Index theorems for quantum graphs

In geometric analysis, an index theorem relates the difference of the numbers
of solutions of two differential equations to the topological structure of the
manifold or bundle concerned, sometimes using the heat kernels of two
higher-order differential operators as an intermediary. In this paper, the case
of quantum graphs is addressed. A quantum graph is a graph considered as a
(singular) one-dimensional variety and equipped with a second-order
differential Hamiltonian H (a "Laplacian") with suitable conditions at
vertices. For the case of scale-invariant vertex conditions (i.e., conditions
that do not mix the values of functions and of their derivatives), the constant
term of the heat-kernel expansion is shown to be proportional to the trace of
the internal scattering matrix of the graph. This observation is placed into
the index-theory context by factoring the Laplacian into two first-order
operators, H =A*A, and relating the constant term to the index of A. An
independent consideration provides an index formula for any differential
operator on a finite quantum graph in terms of the vertex conditions. It is
found also that the algebraic multiplicity of 0 as a root of the secular
determinant of H is the sum of the nullities of A and A*.Comment: 19 pages, Institute of Physics LaTe

### On occurrence of spectral edges for periodic operators inside the Brillouin zone

The article discusses the following frequently arising question on the
spectral structure of periodic operators of mathematical physics (e.g.,
Schroedinger, Maxwell, waveguide operators, etc.). Is it true that one can
obtain the correct spectrum by using the values of the quasimomentum running
over the boundary of the (reduced) Brillouin zone only, rather than the whole
zone? Or, do the edges of the spectrum occur necessarily at the set of
``corner'' high symmetry points? This is known to be true in 1D, while no
apparent reasons exist for this to be happening in higher dimensions. In many
practical cases, though, this appears to be correct, which sometimes leads to
the claims that this is always true. There seems to be no definite answer in
the literature, and one encounters different opinions about this problem in the
community.
In this paper, starting with simple discrete graph operators, we construct a
variety of convincing multiply-periodic examples showing that the spectral
edges might occur deeply inside the Brillouin zone. On the other hand, it is
also shown that in a ``generic'' case, the situation of spectral edges
appearing at high symmetry points is stable under small perturbations. This
explains to some degree why in many (maybe even most) practical cases the
statement still holds.Comment: 25 pages, 10 EPS figures. Typos corrected and a reference added in
the new versio

### Absence of bound states for waveguides in 2D periodic structures

We study a Helmholtz-type spectral problem in a two-dimensional medium
consisting of a fully periodic background structure and a perturbation in form
of a line defect. The defect is aligned along one of the coordinate axes,
periodic in that direction (with the same periodicity as the background), and
bounded in the other direction. This setting models a so-called "soft-wall"
waveguide problem. We show that there are no bound states, i.e., the spectrum
of the operator under study contains no point spectrum.Comment: This is an updated version of our paper (in slightly different form
in Journal of Mathematical Physics). An anonymous reviewer kindly made us
aware that ref. 10 is not applicable in our situation. An application of the
theorem in ref. 10 would have proved the absence of singular continuous
spectrum also. Our result on the absence of point spectrum is not affected by
thi

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