Simplest graphs with small edges: asymptotics for resolvents and holomorphic dependence of spectrum

Abstract

In the work we consider a simplest graph formed by two finite edges and a small edge coupled at a common vertex. The length of the small edge serves as a small parameter. On such graph, we consider the Schrödinger operator with the Kirchoff condition at the internal vertex, the Dirichlet condition on the boundary vertices of finite edges and the Dirichlet or Neumann condition on the boundary vertex of the small edge. We show that such operator converges to a Schrödinger operator on the graph without the small edge in the norm resolvent sense; at the internal vertex one has to impose the Dirichlet condition if the same was on the boundary vertex of the small edge. If the boundary vertex was subject to the Neumann condition, the internal vertex keeps the Kirchoff condition but the coupling constant can change. The main obtained result for the resolvents is the two-terms asymptotics for their resolvents and an estimate for the error term. The second part of the work is devoted to studying the dependence of the eigenvalues on the small parameter. Despite the graph is perturbed singularly, the eigenvalues are holomorphic in the small parameter and are represented by convergent series. We also find out that under the perturbation, there can be stable eigenvalues independent of the parameter. We provide a criterion determining the existence of such eigenvalues. For varying eigenvalues we find the leading terms of their Taylor series.