Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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((\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.

30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
9
The categories of this paper are: nlin
17
The categories of this paper are: q-fin
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
23
The categories of this paper are: solv-int
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th