TY - JOUR
T1 - Analysis of the essential spectrum of singular matrix differential operators
AU - Ibrogimov, O. O.
AU - Siegl, Petr
AU - Tretter, C.
N1 - Funding Information:
The authors thank the referee for valuable comments and they gratefully acknowledge the support of the Swiss National Science Foundation , SNF, grant no. 200020_146477 (O.O. Ibrogimov and C. Tretter) as well as Ambizione grant no. PZ00P2_154786 (P. Siegl). C. Tretter also thanks the Knut och Alice Wallenbergs Stiftelse, Sweden, for a guest professorship 2014/15 and the Matematiska institutionen at Stockholms universitet for the kind hospitality.
Publisher Copyright:
© 2015 Elsevier Inc.
PY - 2016
Y1 - 2016
N2 - 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.
AB - 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.
KW - Essential spectrum
KW - Magnetohydrodynamics
KW - Operator matrix
KW - Schur complement
KW - Stellar equilibrium model
KW - System of singular differential equations
UR - http://www.scopus.com/inward/record.url?scp=84947998771&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2015.10.050
DO - 10.1016/j.jde.2015.10.050
M3 - Article
AN - SCOPUS:84947998771
SN - 0022-0396
VL - 260
SP - 3881
EP - 3926
JO - Journal of Differential Equations
JF - Journal of Differential Equations
IS - 4
ER -