Analysis of the essential spectrum of singular matrix differential operators

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((α, β)) \oplus \bigl(L^2((α, β))\bigr)^n \label{mo} \end{align} singular at $β\in\mathbb R\cup\{\infty\}$ 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" $σ_{\rm ess}^{\rm \,s}(\mathcal A)$ is investigated systematically. Our main results include an explicit description of $σ_{\rm ess}^{\rm \,s}(\mathcal A)$, 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 $π(\cdot,λ)=p-b^*(D-λ)^{-1}b$ of the first Schur complement of $\mathcal A$, a scalar differential operator but non-linear in $λ$; the Nevanlinna behaviour in $λ$ of certain limits $t\!\nearrow\!β$ 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.

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