1,029 research outputs found
Uniqueness in potential scattering with reduced near field data
We consider inverse potential scattering problems where the source of the
incident waves is located on a smooth closed surface outside of the
inhomogeneity of the media. The scattered waves are measured on the same
surface at a fixed value of the energy. We show that this data determines the
bounded potential uniquely.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0374
Wave propagation in periodic networks of thin fibers
We will discuss a one-dimensional approximation for the problem of wave
propagation in networks of thin fibers. The main objective here is to describe
the boundary (gluing) conditions at branching points of the limiting
one-dimensional graph. The results will be applied to Mach-Zehnder
interferometers on chips and to periodic chains of the interferometers. The
latter allows us to find parameters which guarantee the transparency and
slowing down of wave packets
On absence of embedded eigenvalues for Schr\"{o}dinger operators with perturbed periodic potentials
The problem of absence of eigenvalues imbedded into the continuous spectrum
is considered for a Schr\"{o}dinger operator with a periodic potential
perturbed by a sufficiently fast decaying ``impurity'' potential. Results of
this type have previously been known for the one-dimensional case only. Absence
of embedded eigenvalues is shown in dimensions two and three if the
corresponding Fermi surface is irreducible modulo natural symmetries. It is
conjectured that all periodic potentials satisfy this condition. Separable
periodic potentials satisfy it, and hence in dimensions two and three
Schr\"{o}dinger operator with a separable periodic potential perturbed by a
sufficiently fast decaying ``impurity'' potential has no embedded eigenvalues.Comment: LATEX, 15 page
Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem
The paper concerns the isotropic interior transmission eigenvalue (ITE)
problem. This problem is not elliptic, but we show that, using the
Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to
the discreteness of the spectrum as well as to certain results on possible
location of the transmission eigenvalues. If the index of refraction
is real, we get a result on the existence of infinitely many
positive ITEs and the Weyl type lower bound on its counting function. All the
results are obtained under the assumption that does not vanish at the
boundary of the obstacle or it vanishes identically, but its normal derivative
does not vanish at the boundary. We consider the classical transmission problem
as well as the case when the inhomogeneous medium contains an obstacle. Some
results on the discreteness and localization of the spectrum are obtained for
complex valued .Comment: A small correction is made in formulas (11), (12) after the paper was
published in "Inverse Problems", 29, 201
Solution of the initial value problem for the focusing Davey-Stewartson II system
We consider a focusing Davey-Stewartson system and construct the solution of
the Cauchy problem in the possible presence of exceptional points (and/or
curves)
Radiation Conditions for the Difference Schr\"{o}dinger Operators
The problem of determining a unique solution of the Schr\"{o}dinger equation
on the lattice is
considered, where is the difference Laplacian and both and
have finite supports It is shown that there is an exceptional set of
points on for which the limiting absorption principle
fails, even for unperturbed operator (). This exceptional set consists
of the points when is even and when is odd. For all values of the radiation conditions are found which single
out the same solutions of the problem as the ones determined by the limiting
absorption principle. These solutions are combinations of several waves
propagating with different frequencies, and the number of waves depends on the
value of $\lambda.
Laplace Operator in Networks of Thin Fibers: Spectrum Near the Threshold
Our talk at Lisbon SAMP conference was based mainly on our recent results
(published in Comm. Math. Phys.) on small diameter asymptotics for solutions of
the Helmgoltz equation in networks of thin fibers. The present paper contains a
detailed review of these results under some assumptions which make them much
more transparent. It also contains several new theorems on the structure of the
spectrum near the threshold. small diameter asymptotics of the resolvent, and
solutions of the evolution equation
Transition from a network of thin fibers to the quantum graph: an explicitly solvable model
We consider an explicitly solvable model (formulated in the Riemannian
geometry terms) for a stationary wave process in a specific thin domain with
the Dirichlet boundary conditions on the boundary of the domain. The transition
from the solutions of the scattering problem to the solutions of a problem on
the limiting quantum graph is studied. We calculate the Lagrangian gluing
conditions at vertices for the problem 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 of a
problem in a neighborhood of each vertex. Near the bottom of the absolutely
continuous spectrum the wave propagation is generically suppressed, and the
gluing condition is degenerate (any solution of the limiting problem is zero at
each vertex)
Large time behavior of the solutions to the difference wave operators
The Cauchy problem for two dimensional difference wave operators is
considered with potentials and initial data supported in a bounded region. The
large time asymptotic behavior of solutions is obtained. In contrast to the
continuous case (when the problem in the Euclidian space is considered, not on
the lattice) the resolvent of the corresponding stationary problem has
singularities on the continuous spectrum, and they contribute to the
asymptotics
Non-random perturbations of the Anderson Hamiltonian
The Anderson Hamiltonian is considered, where
is a random potential of Bernoulli type. The operator is perturbed by a
non-random, continuous potential , decaying at infinity. It will
be shown that the borderline between finitely, and infinitely many negative
eigenvalues of the perturbed operator, is achieved with a decay of the
potential as .Comment: Minor changes in the introductory section
- β¦