Analysis of the essential spectrum of singular matrix differential operators

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.