Abstract Wave Equations and Associated Dirac-Type Operators

TL;DR

This paper explores the deep mathematical relationship between abstract damped wave equations and Dirac-type operators, analyzing spectral properties, conserved quantities, and energy distribution, especially when zero is in the continuous spectrum.

Contribution

It establishes unitary equivalence between wave equation generators and Dirac-type operators, and provides detailed spectral and domain analyses, including special cases with damping and spectral edge conditions.

Findings

Unitary equivalence between wave generators and Dirac operators.

Explicit computation of semigroup growth bounds.

Analysis of energy equipartition and spectral properties.

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 $A^*A$. 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 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 ($R \neq 0$) abstract wave equations as well as the cases $A^* A \geq \epsilon I_{\mathcal{H}_1}$ for some $\epsilon > 0$ and $0 \in \sigma (A^* A)$ (but 0 not an eigenvalue of $A^*A$) are separately studied in detail.