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Abstract wave equations and associated Dirac-type operators

Content Provider	Springer Nature Link
Author	Gesztesy, Fritz Holden, Helge Goldstein, Jerome A. Teschl, Gerald
Copyright Year	2011
Abstract	We discuss the unitary equivalence of generators G A,R associated with abstract damped wave equations of the type $${\ddot{u} + R \dot{u} + A^A u = 0}$$ in some Hilbert space $${\mathcal{H}_1}$$ and certain non-self-adjoint Dirac-type operators Q A,R (away from the nullspace of the latter) in $${\mathcal{H}_1 \oplus \mathcal{H}_2}$$ . The operator Q A,R represents a non-self-adjoint perturbation of a supersymmetric self-adjoint Dirac-type operator. Special emphasis is devoted to the case where 0 belongs to the continuous spectrum of AA. In addition to the unitary equivalence results concerning G A,R and Q A,R , we provide a detailed study of the domain of the generator G A,R , consider spectral properties of the underlying quadratic operator pencil $${M(z) = \|A\|^2 - iz R - z^2 I_{\mathcal{H}_1}, z\in\mathbb{C}}$$ , derive a family of conserved quantities for abstract wave equations in the absence of damping, and prove equipartition of energy for supersymmetric self-adjoint Dirac-type operators. The special example where R represents an appropriate function of \|A\| is treated in depth, and the semigroup growth bound for this example is explicitly computed and shown to coincide with the corresponding spectral bound for the underlying generator and also with that of the corresponding Dirac-type operator. The cases of undamped (R = 0) and damped (R ≠ 0) abstract wave equations as well as the cases $${A^* A \geq \varepsilon I_{\mathcal{H}_1}}$$ for some $${\varepsilon > 0}$$ and $${0 \in \sigma (A^* A)}$$ (but 0 not an eigenvalue of A*A) are separately studied in detail.
Ending Page	676
Page Count	46
Starting Page	631
File Format	PDF
ISSN	03733114
e-ISSN	16181891
Journal	Annali di Matematica Pura ed Applicata
Issue Number	4
Volume Number	191
Language	English
Publisher	Springer-Verlag
Publisher Date	2011-04-10
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Wave equation Supersymmetry Partial differential operators Boundary value problems for second-order elliptic equations Boundary value problems for higher-order elliptic equations Wave equations Damping terms General (adjoints, conjugates, products, inverses, domains, ranges, etc.) Mathematics Quadratic operator pencils General topics in linear spectral theory Initial value problems for second-order hyperbolic equations Spectrum, resolvent Semigroups Dirac operators
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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