A general approximation of quantum graph vertex couplings by scaled Schroedinger operators on thin branched manifolds

TL;DR

This paper presents a method to approximate all self-adjoint vertex couplings in quantum graphs using scaled magnetic Schrödinger operators on thin branched manifolds, with proven norm-resolvent convergence as the tubes shrink.

Contribution

It introduces a general approximation scheme for quantum graph vertex couplings via scaled Schrödinger operators on manifolds, extending previous specific cases.

Findings

Achieved norm-resolvent convergence of the approximations

Developed a local topology change technique near vertices

Extended approximation methods to manifolds with boundaries

Abstract

We demonstrate that any self-adjoint coupling in a quantum graph vertex can be approximated by a family of magnetic Schroedinger operators on a tubular network built over the graph. If such a manifold has a boundary, Neumann conditions are imposed at it. The procedure involves a local change of graph topology in the vicinity of the vertex; the approximation scheme constructed on the graph is subsequently `lifted' to the manifold. For the corresponding operator a norm-resolvent convergence is proved, with the natural identification map, as the tube diameters tend to zero.