Non-self-adjoint Dirac operators on graphs

TL;DR

This paper introduces and analyzes non-self-adjoint Dirac operators on finite metric graphs, deriving spectral properties and symmetry conditions using boundary triple methods, with implications for quantum graph models.

Contribution

It develops a framework for non-self-adjoint Dirac operators on graphs, including spectral analysis and symmetry criteria, expanding understanding beyond self-adjoint cases.

Findings

Eigenvalue behavior varies with graph structure

Conditions for symmetry under physical transformations

A variant of the Birman-Schwinger principle for these operators

Abstract

In this paper we introduce and study generally non-self-adjoint realizations of the Dirac operator on an arbitrary finite metric graph. Employing the robust boundary triple framework, we derive, in particular, a variant of the Birman Schwinger principle for its eigenvalues, and with an example of a star shaped graph we show that the point spectrum may exhibit diverse behaviour. Subsequently, we find sufficient and necessary conditions on transmission conditions at the graph's vertices under which the Dirac operator on the graph is symmetric with respect to the parity, the time reversal, or the charge conjugation transformation.