Analysis of the essential spectrum of singular matrix differential operators

TL;DR

This paper provides a comprehensive analysis of the essential spectrum of singular matrix differential operators, including explicit descriptions, criteria for spectral components, and applications to symmetric stellar models.

Contribution

It introduces a systematic approach to characterize the singular part of the essential spectrum for a class of matrix differential operators, including criteria for its presence and topological analysis.

Findings

Explicit description of the singular part of the essential spectrum.

Criteria for the absence or presence of the spectrum component.

Application to symmetric stellar equilibrium models.

Abstract

A complete analysis of the essential spectrum of matrix-differential operators $\mathcal A$ of the form \begin{align} \begin{pmatrix} -\displaystyle{\frac{\rm d}{\rm d t}} p \displaystyle{\frac{\rm d}{\rm d t}} + q & -\displaystyle{\frac{\rm d}{\rm d t}} b^* \! + c^* \\[2mm] \hspace{6mm} b \displaystyle{\frac{\rm d}{\rm d t}} + c & \hspace{4mm} D \end{pmatrix} \quad \text{in } \ L^2((\alpha, \beta)) \oplus \bigl(L^2((\alpha, \beta))\bigr)^n \label{mo} \end{align} singular at $\beta\in\mathbb R\cup\{\infty\}$ is given; the coefficient functions $p$, $q$ are scalar real-valued with $p>0$, $b$, $c$ are vector-valued, and $D$ is Hermitian matrix-valued. The so-called "singular part of the essential spectrum" $\sigma_{\rm ess}^{\rm \,s}(\mathcal A)$ is investigated systematically. Our main results include an explicit description of $\sigma_{\rm ess}^{\rm \,s}(\mathcal A)$, criteria for its absence and presence; an analysis of its topological structure and of the essential spectral radius. Our key tools are: the asymptotics of the leading coefficient $\pi(\cdot,\lambda)=p-b^*(D-\lambda)^{-1}b$ of the first Schur complement of $\mathcal A$, a scalar differential operator but non-linear in $\lambda$; the Nevanlinna behaviour in $\lambda$ of certain limits $t\!\nearrow\!\beta$ of functions formed out of the coefficients in $\mathcal A$. The efficacy of our results is demonstrated by several applications; in particular, we prove a conjecture on the essential spectrum of some symmetric stellar equilibrium models.