Approximation of quantum graph vertex couplings by scaled Schr\"odinger operators on thin branched manifolds

TL;DR

This paper demonstrates how Schrödinger operators on thin branched manifolds with tailored potentials can approximate various quantum graph vertex couplings, including delta and delta'-type interactions, expanding the understanding of quantum graph approximations.

Contribution

It introduces a method to approximate complex quantum graph vertex couplings using scaled Schrödinger operators on thin manifolds, including those with discontinuous wavefunctions.

Findings

Schrödinger operators with potentials approximate delta and delta'-couplings

The approach works for symmetric delta'-couplings

Conjecture: method applies to all time-reversal invariant couplings

Abstract

We discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds. We show that if a Neumann type Laplacian on such manifolds is amended by suitable potentials, the resulting Schr\"odinger operators can approximate non-trivial vertex couplings. The latter include not only the delta-couplings but also those with wavefunctions discontinuous at the vertex. We work out the example of the symmetric delta'-couplings and conjecture that the same method can be applied to all couplings invariant with respect to the time reversal.