Approximations of permutation-symmetric vertex couplings in quantum graphs

TL;DR

This paper studies how to approximate permutation-symmetric vertex couplings in quantum graphs using simpler models with delta interactions and point interactions, providing a way to simulate complex boundary conditions.

Contribution

It introduces a method to approximate permutation-symmetric vertex couplings in quantum graphs via norm-resolvent convergence with simpler operators involving delta and point interactions.

Findings

Operators can be approximated in the norm-resolvent sense

Approximation involves delta coupling and point interactions

Provides a practical approach for modeling complex boundary conditions

Abstract

We consider boundary conditions at the vertex of a star graph which make Schroedinger operators on the graph self-adjoint, in particular, the two-parameter family of such conditions invariant with respect to permutations of graph edges. It is proved that the corresponding operators can be approximated in the norm-resolvent sense by elements of another Schroedinger operator family on the same graph in which the delta coupling is imposed at the vertex and an additional point interaction is placed at each edge provided the coupling parameters are properly chosen.