Non-self-adjoint Dirac operators on graphs

TL;DR

This work develops a robust framework for non-self-adjoint Dirac operators on finite metric graphs using boundary triples, yielding a Birman–Schwinger type eigenvalue principle and a Krein resolvent formula for the graph operator $D^{\Lambda_{A,B}}$. It provides a concrete edge-by-edge construction, builds the graph operator via a direct-sum approach, and demonstrates spectral behavior through a star-graph example. The authors also establish necessary and sufficient conditions for parity, time-reversal, and charge-conjugation symmetries in terms of the transmission data $(A,B)$, with explicit simplifications in the star-graph setting. Overall, the paper extends non-self-adjoint boundary conditions to relativistic Dirac operators on graphs, showing how boundary data govern both spectral properties and symmetry behaviors, and highlights the stability of essential spectra under local transmission conditions.

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.

Figures (1)

• Figure 1: A graph with three vertices $(v_1,v_2,v_3)$ and four edges $(j_1,\ldots,j_4)$ whose incidence matrix $G$ is given in \ref{['eq:incidence_ex']}.

Theorems & Definitions (80)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3: BrGePaBT_2008
• Theorem 2.4
• Theorem 2.5
• Proposition 3.1
• proof
• Remark 3.2
• Proposition 3.3
• proof