Perturbation of Threshold of the Essential Spectrum of the Schrödinger Operator on the Simplest Graph with a Small Edge

Resumo

On a star graph consisting of two infinite edges and one small edge, we consider the Schrödinger operators with piecewise-constant potentials on the infinite edges and with a singular potential on the small edge respectively. A δ′-interaction is given at the interior vertex of the graph, and the Dirichlet or Neumann condition is imposed at the boundary vertex of the small edge. We determine the limit boundary conditions, obtain two-term asymptotics for the resolvents in the operator norm and error estimates. The phenomenon of isolated eigenvalues emerging from the threshold of the essential spectrum is discussed. We establish efficient and easily verified sufficient conditions for the existence or absence of such eigenvalues. We establish the holomorphic dependence of the appeared eigenvalues on the edge length and write explicitly the first terms of the Taylor expansions of such eigenvalues.