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Our talk at Lisbon SAMP conference was based mainly on our recent results on small diameter asymptotics for solutions of the Helmgoltz equation in networks of thin fibers. These results were published in [21]. The present paper contains a detailed review of [21] under some assumptions which make the results much more transparent. It also contains several(More)
We consider 1-D Schrr odinger operators on L 2 (R +) with slowly decaying potentials. Under some conditions on the potential, related to the rst integrals of the KdV equation, we prove that the a.c. spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable. Examples show that for special classes of sparse potentials(More)
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)
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)
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 Γ. We calculate the Lagrangian gluing conditions at vertices v ∈ Γ for the problems on the limiting graph. If the frequency of the incident wave is above the bottom(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)