Scattering problems for symmetric systems with dissipative boundary conditions

TL;DR

This paper investigates the scattering behavior of symmetric systems with dissipative boundary conditions, focusing on the properties of solutions, wave operators, and inverse scattering problems, especially in Maxwell systems with specific boundary conditions.

Contribution

It provides a new analysis of the completeness of wave operators and characterizes the scattering kernel for such systems, including Maxwell equations with special boundary conditions.

Findings

Wave operators are not complete if asymptotically disappearing solutions exist.

Representation of the scattering kernel is obtained.

Inverse back-scattering problem related to the leading term is examined.

Abstract

We study symmetric systems with dissipative boundary conditions. The solutions of the mixed problems for such systems are given by a contraction semigroup $V(t)f = e^{tG_b}f, t \geq 0$. The solutions $u(t, x) = V(t)f$, where $f$ is an eigenfunction of the generator $G_b$ with eigenvalue $\lambda,\Re \lambda < 0,$ are called asymptotically disappearing (ADS). We prove that the wave operators are not complete if there exist (ADS). This is the case for Maxwell system with special boundary conditions in the exterior of the sphere. We obtain a representation of the scattering kernel and we examine the inverse back-scattering problem related to the leading term of the scattering kernel.