Approximation of a general singular vertex coupling in quantum graphs

TL;DR

This paper presents a method to approximate any singular vertex coupling in quantum graphs by using a construction with decoupled edges, delta potentials, and vector potentials, achieving convergence in the norm-resolvent sense.

Contribution

It provides a universal approximation scheme for all singular vertex couplings in quantum graphs, solving a longstanding open problem.

Findings

Any prescribed singular vertex coupling can be approximated by the proposed construction.

The approximation converges in the norm-resolvent sense as connecting edge lengths shrink.

The method involves delta potentials and vector potentials on decoupled edges.

Abstract

The longstanding open problem of approximating all singular vertex couplings in a quantum graph is solved. We present a construction in which the edges are decoupled; an each pair of their endpoints is joined by an edge carrying a $\delta$ potential and a vector potential coupled to the "loose" edges by a $\delta$ coupling. It is shown that if the lengths of the connecting edges shrink to zero and the potentials are properly scaled, the limit can yield any prescribed singular vertex coupling, and moreover, that such an approximation converges in the norm-resolvent sense.