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We discuss the potential scattering on the noncompact star graph. The Schrödinger operator with the short-range potential localizing in a neighborhood of the graph vertex is considered. We study the asymptotic behavior the corresponding scattering matrix in the zero-range limit. It has been known for a long time that in dimension 1 there is no non-trivial Hamiltonian with the distributional potential δ ′ , i.e., the δ ′ potential acts as a totally reflecting wall. Several authors have, in recent years, studied the scattering properties of the regularizing potentials αε −2 Q(x/ε) approximating the first derivative of the Dirac delta function. A non-zero transmission through the regularized potential has been shown to exist as ε → 0. We extend these results to star graphs with the point interaction, which is an analogue of δ ′ potential on the line. We prove that generically such a potential on the graph is opaque. We also show that there exists a countable set of resonant intensities for which a partial transmission through the potential occurs. This set of resonances is referred to as the resonant set and is determined as the spectrum of an auxiliary Sturm-Liouville problem associated with Q on the graph.

We study Schrödinger operators on star metric graphs with potentials of the form αε −2 Q(ε −1 x). In dimension 1 such potentials, with additional assumptions on Q, approximate in the sense of distributions as ε → 0 the first derivative of the Dirac delta-function. We establish the convergence of the Schrödinger operators in the uniform resolvent topology and show that the limit operator depends on α and Q in a very nontrivial way.

Abstract.In this paper, we study the inverse scattering problem for ZS-AKNS systems on the half-line with general boundary conditions at the origin. For the class of potentials with certain integrability properties, we give a complete description of the corresponding scattering functions S, justify the algorithm reconstructing the potential and the boundary conditions from S, and prove that the scattering map is homeomorphic.Mathematics Subject Classification. Primary 34L25; Secondary 34L40, 81U20, 81U40.

In this paper, we study the inverse scattering problem for energy-dependent Schrödinger equations on the half-line with energy-dependent boundary conditions at the origin. Under certain positivity and very mild regularity assumptions, we transform this scattering problem to the one for non-canonical Dirac systems and show that, in turn, the latter can be placed within the known scattering theory for ZS-AKNS systems. This allows us to give a complete description of the corresponding scattering functions S for the class of problems under consideration and justify an algorithm of reconstructing the problem from S.