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We consider an explicitly solvable model (formulated in the Rie-mannian geometry terms) for a stationary wave process in a specific thin domain Ωε with the Dirichlet boundary conditions on ∂Ωε. The transition from the solutions of the scattering problem on Ωε to the solutions of a problem on the limiting quantum graph Γ is studied. We calculate the… (More)

- Valery Imaikin, Alexander Komech, Boris Vainberg
- 2008

We establish the long time soliton asymptotics for the translation invariant nonlinear system consisting of the Klein-Gordon equation coupled to a charged relativistic particle. The coupled system has a six dimensional invariant manifold of the soliton solutions. We show that in the large time approximation any finite energy solution, with the initial state… (More)

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… (More)

The problem of absence of eigenvalues imbedded into the continuous spectrum is considered for a Schrödinger 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… (More)

We consider the interior transmission eigenvalue (ITE) problem, which arises when scattering by inhomogeneous media is studied. The ITE problem is not self-adjoint. We show that positive ITEs are observable together with plus or minus signs that are defined by the direction of motion of the corresponding eigenvalues of the scattering matrix (when the latter… (More)

We consider the Helmholtz equation in the half space and suggest two methods for determining the boundary impedance from knowledge of the far field pattern of the time-harmonic incident wave. We introduce a potential for which the far field patterns in specially selected directions represent its Fourier coefficients. The boundary impedance is then… (More)

- Yuri A Godin, Boris Vainberg
- 2008

The inverse scattering problem of determining the boundary impedance from knowledge of the time harmonic incident wave and the far-field pattern of the scattered wave is considered. We use the finite-difference approximation for the Helmholtz equation along with the exact radiation condition for the discrete equation. The approach involves two steps. First,… (More)