 10204 Fritz Gesztesy, Jerome A. Goldstein, Helge Holden, and Gerald Teschl
 Abstract Wave Equations and Associated DiracType Operators
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Dec 22, 10

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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 nonselfadjoint Diractype
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 nonselfadjoint perturbation of
a supersymmetric selfadjoint Diractype operator. Special emphasis is devoted to the case where $0$ belongs to the continuous spectrum of $A^*A$.
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 selfadjoint
Diractype 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 Diractype
operator.
The cases of undamped ($R=0$) and damped ($R
eq 0$) abstract wave equations as well as the cases $A^* A \geq arepsilon I_{\mathcal{H}_1}$ for some
$arepsilon > 0$ and $0 \in \sigma (A^* A)$ (but $0$ not an eigenvalue of
$A^*A$) are separately studied in detail.
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