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Analysis of the essential spectrum of singular matrix differential operators
Stockholm University, Faculty of Science, Department of Mathematics. Universit├Ąt Bern, Switzerland.
Number of Authors: 32016 (English)In: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 260, no 4, p. 3881-3926Article in journal (Refereed) Published
Abstract [en]

A complete analysis of the essential spectrum of matrix-differential operators A of the form (-d/dt p d/dt + q -d/dt b* + c*) b d/dt + c D) in L2((alpha, beta)) circle times (L-2((alpha, beta)))(n) singular at beta is an element of R boolean OR {infinity} 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(s)(ess)(A) is investigated systematically. Our main results include an explicit description of sigma(s)(ess)(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(center dot, lambda) = p - b* (D - lambda)(-1) b of the first Schur complement of (0.1), a scalar differential operator but non-linear in lambda; the Nevanlinna behaviour in lambda of certain limits t NE arrow beta 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.

Place, publisher, year, edition, pages
2016. Vol. 260, no 4, p. 3881-3926
Keywords [en]
Essential spectrum, System of singular differential equations, Operator matrix, Schur complement, Magnetohydrodynamics, Stellar equilibrium model
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-175574DOI: 10.1016/j.jde.2015.10.050ISI: 000373536700024OAI: oai:DiVA.org:su-175574DiVA, id: diva2:1367728
Available from: 2019-11-05 Created: 2019-11-05 Last updated: 2019-11-05Bibliographically approved

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