Spectral theory of some matrix differential operators of mixed order

Konstantinov, A. Yu.

doi:10.1007/bf02513093

Cited by 8 publications

(4 citation statements)

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“…Operator matrices with this particular structure appear in several problems in mathematical physics, see e.g. [8,20,21,22,26,29]. Due to the apparent independence of their entries, however, their spectral analysis is not straightforward and previous results typically rely on the regularity or special form of coefficients.…”

Section: Singular Coefficient Matrix Differential Operatorsmentioning

confidence: 99%

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Schur complement dominant operator matrices

Gerhat¹

2022

Preprint

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We propose a method for the spectral analysis of unbounded operator matrices in a general setting which fully abstains from standard perturbative arguments. Rather than requiring the matrix to act in a Hilbert space H, we extend its action to a suitable distributional triple D ⊂ H ⊂ D − and restrict it to its maximal domain in H. The crucial point in our approach is the choice of the spaces D and D − which are essentially determined by the Schur complement of the matrix. We show spectral equivalence between the resulting operator matrix in H and its Schur complement, which allows to pass from a suitable representation of the Schur complement (e.g. by generalised form methods) to a representation of the operator matrix. We thereby generalise classical spectral equivalence results imposing standard dominance patterns.The abstract results are applied to damped wave equations with possibly unbounded and/or singular damping, to Dirac operators with Coulomb-type potentials, as well as to generic second order matrix differential operators. By means of our methods, previous regularity assumptions can be weakened substantially.

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Section: Singular Coefficient Matrix Differential Operatorsmentioning

confidence: 99%

“…Problems of this type arise in areas like magnetohydrodynamics or astrophysics and have been previously studied in e.g. [8,20,21,22,26,29]. Our methods allow us to avoid typical technical assumptions like q ∈ C(Ω), b, c ∈ C 1 (Ω) n and d ∈ C 1 (Ω), see e.g [21].…”

Section: Introductionmentioning

confidence: 99%

Schur complement dominant operator matrices

Gerhat¹

2022

Preprint

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“…The study of the block operator matrix is the subject of many authors under different assumptions. In this direction some issues may be found in the literature, we can quote for example [1,2,10,16,24]. Recently, an account research and a wide panorama of methods to investigate the spectral theory of block operator matrices is given in [3,4,5,11,14,25].…”

Section: Introductionmentioning

confidence: 99%

On Relative Essential Spectra of Block Operator Matrices and Application

Charfi¹,

Walha²

2016

Bulletin of the Korean Mathematical Society

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“…Furthermore, in , we find that an account research and a wide panorama of methods to investigate the spectral theory of block operator matrices are given. More precisely, the description of the Wolf essential spectrum of a block operator matrix

scriptA

is investigated in ,Theorem 2.4.8, by C. Tretter to improve and generalize some results given by A. Y. Konstantinov in ,Theorem 1 for self‐adjoint block operator matrices in Hilbert spaces. In ,Theorem 2.4.8, it is assumed that the domains of entries operators satisfy