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 $\sqrt{n(x)}$ 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 $n(x)-1$ 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 $n(x)$.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 $\left(\Delta+q-\lambda\right) \psi=f$ on the lattice $\mathbb{Z}^{d}$ is considered, where $\Delta$ is the difference Laplacian and both $f$ and $q$ have finite supports$.$ It is shown that there is an exceptional set $S_{0}$ of points on $Sp(\Delta)=[-2d,2d]$ for which the limiting absorption principle fails, even for unperturbed operator ($q(x)=0$). This exceptional set consists of the points $\left\{ \pm4n\right\}$ when $d$ is even and $\left\{ \pm2(2n+1)\right\}$ when $d$ is odd. For all values of $\lambda \in[-2d,2d]\backslash S_{0},$ 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 $H_0=-\Delta+V(x,\omega)$ is considered, where $V$ is a random potential of Bernoulli type. The operator $H_0$ is perturbed by a non-random, continuous potential $-w(x) \leq 0$, 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 $-w(x)$ as $O(\ln^{-2/d} |x|)$.Comment: Minor changes in the introductory section