Dirac-Krein systems on star graphs

TL;DR

This paper analyzes the spectral properties of Dirac-Krein operators on star graphs, deriving resolvent formulas, eigenvalue interlacing, and trace formulas, with focus on Robin boundary conditions and asymptotic eigenvalue distribution.

Contribution

It introduces a comprehensive spectral analysis framework for Dirac-Krein systems on star graphs, including resolvent formulas, Weyl functions, and eigenvalue asymptotics, with novel insights into eigenvalue interlacing and dislocation index.

Findings

Derived Krein's resolvent formula for the operator

Studied eigenvalue interlacing and multiplicities

Established a trace formula and eigenvalue asymptotics

Abstract

We study the spectrum of a self-adjoint Dirac-Krein operator with potential on a compact star graph $\mathcal G$ with a finite number $n$ of edges. This operator is defined by a Dirac-Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of $\mathcal G$. Special attention is paid to Robin matching conditions with parameter $\tau \in\mathbb R\cup\{\infty\}$. Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Krein's resolvent formula, introduce corresponding Weyl-Titchmarsh functions, study the multiplicities, dependence on $\tau$, and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for $R\to \infty$, the difference of the number of eigenvalues in the intervals $[0,R)$ and $[-R,0)$ deviates from some integer $\kappa_0$, which we call dislocation index, at most by $n+2$.