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.
Original language | English |
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Pages (from-to) | 3881-3926 |
Number of pages | 46 |
Journal | Journal of Differential Equations |
Volume | 260 |
Issue number | 4 |
Early online date | 28 Nov 2015 |
DOIs | |
Publication status | Published - 01 Jan 2016 |
Externally published | Yes |
Keywords
- Essential spectrum
- Magnetohydrodynamics
- Operator matrix
- Schur complement
- Stellar equilibrium model
- System of singular differential equations
ASJC Scopus subject areas
- Analysis