Pavel Kurasov

On the inverse scattering problem on branching graphs

The inverse scattering problem on branching graphs is studied. The definition of the Schrodinger operator on such graphs is discussed. The operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices. It is shown that in general the scattering matrix does not determine the topology of the graph, the potentials on the edges and the boundary conditions uniquely.

Inverse spectral problem for quantum graphs

The inverse spectral problem for the Laplace operator on a finite metric graph is investigated. It is shown that this problem has a unique solution for graphs with rationally independent edges and without vertices having valence 2. To prove the result, a trace formula connecting the spectrum of the Laplace operator with the set of periodic orbits for the metric graph is established.

Symmetries of Schrödinger operator with point interactions

The transformations of all the Schrödinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling Wλ are presented. In particular, those operators which are invariant (possibly up to a scale) are selected. Some recent papers on related topics are commented upon.

Point Interactions: PT -Hermiticity and Reality of the Spectrum

AbstractGeneral point interactions for the second derivative operator in one dimension are studied. In particular,

On the Inverse Problem for the Rational Reflection Coefficient

\mathcal{P}\mathcal{T}

Finite rank singular perturbations and distributions with discontinuous testfunctions

-self-adjoint point interactions with the support at the origin and at points ±l are considered. The spectrum of such non-Hermitian operators is investigated and conditions when the spectrum is pure real are presented. The results are compared with those for standard self-adjoint point interactions.

Spectral gap for quantum graphs and their edge connectivity

Rank one perturbations of selfadjoint operators which are not necessarily semibounded are studied in the present paper. It is proven that such perturbations are uniquely defined, if they are bounded in the sense of forms. We also show that form unbounded rank one perturbations can be uniquely defined if the original operator and the perturbation are homogeneous with respect to a certain one parameter semigroup. The perturbed operator is defined using the extension theory for symmetric operators. The resolvent of the perturbed operator is calculated using Kreins formula. It is proven that every rank one perturbation can be approximated in the operator norm. We prove that some form unbounded perturbations can be approximated in the strong resolvent sense without renormalization of the coupling constant only if the original operator is not semibounded. The present approach is applied to study first derivative and Dirac operators with point interaction, in one dimension.

Inverse problems for Aharonov-Bohm rings

The inverse scattering problem on the half-axis for long range potentials is studied. It is shown that the solution of the inverse problem contains arbitrary real parameters iven if no bound states are present. Connections with the inverse problem on the whole axis are discussed.