Locally finite extensions and Gesztesy-\v{S}eba realizations for the Dirac operator on a metric graph

TL;DR

This paper investigates self-adjoint extensions of symmetric operators on metric graphs, focusing on Dirac operators with point interactions, and provides conditions for spectral properties, including cases with arbitrarily small edge lengths.

Contribution

It introduces a framework for analyzing locally finite extensions of Dirac operators on metric graphs using boundary triplets, even with small edge lengths, advancing spectral theory in this context.

Findings

Established conditions for self-adjointness and semi-boundedness.

Derived criteria for spectrum discreteness.

Applied results to Dirac operators with point interactions on graphs.

Abstract

We study extensions of direct sums of symmetric operators $S=\oplus_{n\in\mathbb{N}} S_n$. In general there is no natural boundary triplet for $S^*$ even if there is one for every $S_n^*$, $n\in\mathbb{N}$. We consider a subclass of extensions of $S$ which can be described in terms of the boundary triplets of $S_n^*$ and investigate the self-adjointness, the semi-boundedness from below and the discreteness of the spectrum. Sufficient conditions for these properties are obtained from recent results on weighted discrete Laplacians. The results are applied to Dirac operators on metric graphs with point interactions at the vertices. In particular, we allow graphs with arbitrarily small edge length.