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Systems of Singular Differential Operators of Mixed Order and Applications to 1 ‐ Dimensional MHD Problems
Author(s) -
Hardt Volker,
Mennicken Reinhard,
Naboko Serguei
Publication year - 1999
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.3212050103
Subject(s) - mathematics , gravitational singularity , singularity , perturbation (astronomy) , differential operator , spectrum (functional analysis) , essential spectrum , mathematical analysis , operator (biology) , pure mathematics , ordinary differential equation , extension (predicate logic) , singular perturbation , mathematical physics , differential equation , physics , quantum mechanics , biochemistry , chemistry , repressor , computer science , transcription factor , gene , programming language
In some weighted L 2 vector space we study a symmetric semibounded operator IL' 0 which is given by a 3 x 3 system of ordinary differential operators on an intervall [0, r o ] with a singularity at r = 0 (see (0.1)). This system can be considered as a “smooth” perturbation of a more specific physical model describing the oscillations of plasma in an equilibrium configuration in a cylindrical domain (see (1.12)). This perturbation is smooth in the sense that in the system under study in comparison with the physical model only the smooth parts of the coefficients are changed conserving all types of singularities. It is the goal of this paper to construct a suitable selfadjoint extension IL of the symmetric operator IL' 0 (and its closure IL 0 ) and to determine the essential spectrum of this extension. The essential spectrum consists of two bands (which may overlap) if we exclude the singularities by considering the system on an interval ( r 1,ro] with 0 < r 1 < r o. In the corresponding physical model these bands are called Alfén spectrum and slow magnetosonic spectrum. It is shown that the singularity in 0 generates additional components of the essential spectrum which under specific conditions, as in the case of the physical model, “disappear” in the two bands known from the “regular” case (r1, ro) with r 1 → 0.