Approximations of singular vertex couplings in quantum graphs

TL;DR

This paper explores how to approximate complex singular vertex couplings in quantum graphs, especially when wave functions are discontinuous, by using scaled delta interactions and extended graph structures.

Contribution

It introduces a method to approximate a broad class of singular vertex couplings in quantum graphs via scaled delta interactions and additional edges, expanding the understanding of boundary condition approximations.

Findings

Cheon-Shigehara technique yields a 2n-parameter family of boundary conditions.

Using extra edges, all time-reversal invariant couplings can be approximated.

Approximations are achieved in the norm resolvent topology.

Abstract

We discuss approximations of the vertex coupling on a star-shaped quantum graph of $n$ edges in the singular case when the wave functions are not continuous at the vertex and no edge-permutation symmetry is present. It is shown that the Cheon-Shigehara technique using $\delta$ interactions with nonlinearly scaled couplings yields a $2n$-parameter family of boundary conditions in the sense of norm resolvent topology. Moreover, using graphs with additional edges one can approximate the ${n+1\choose 2}$-parameter family of all time-reversal invariant couplings.