Non-compact quantum graphs with summable matrix potentials

TL;DR

This paper analyzes the spectral properties of noncompact quantum graphs with summable matrix potentials, showing the absence of singular continuous spectrum, conditions for absolute continuity, and deriving a Bargmann-type estimate for negative eigenvalues.

Contribution

It introduces a comprehensive spectral analysis of Hamiltonians on noncompact quantum graphs with summable matrix potentials, including spectrum characterization and scattering matrix formulas.

Findings

Singular continuous spectrum is empty for the studied Hamiltonians.

Conditions for pure absolute continuity of the positive spectrum are provided.

A Bargmann-type estimate for negative eigenvalues is established.

Abstract

Let $\mathcal{G}$ be a metric noncompact connected graph with finitely many edges. The main object of the paper is the Hamiltonian ${\bf H}_{\alpha}$ associated in $L^2(\mathcal{G};\mathbb{C}^m)$ with a matrix Sturm-Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian ${\bf H}_{\alpha}$ as well as any other self-adjoint realization of the Sturm-Liouville expression is empty. We also indicate conditions on the graph ensuring pure absolute continuity of the positive part of ${\bf H}_{\alpha}$. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of ${\bf H}_{\alpha}$ is obtained. Additionally, for a star graph $\mathcal{G}$ a formula is found for the scattering matrix of the pair $\{{\bf H}_{\alpha}, {\bf H}_D\}$, where ${\bf H}_D$ is the Dirichlet operator on $\mathcal{G}$.