Analysis of the essential spectrum of singular matrix differential operators

Abstract

A complete analysis of the essential spectrum of matrix-differential operators A of the form(0.1)(−ddtpddt+q−ddtb⁎+c⁎bddt+cD)in L2((α,β))⊕(L2((α,β)))n singular at β∈R∪{∞} 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” σesss(A) is investigated systematically. Our main results include an explicit description of σesss(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(⋅,λ)=p−b⁎(D−λ)−1b of the first Schur complement of (0.1), a scalar differential operator but non-linear in λ; the Nevanlinna behaviour in λ of certain limits t↗β of functions formed out of the coefficients in (0.1). 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.