A necessary and sufficient condition on scattering for the regularly hyperbolic systems

TL;DR

This paper establishes a precise criterion for scattering in regularly hyperbolic systems with time-dependent coefficients, linking asymptotic freedom to the stability of coefficients, and constructs the scattering operator.

Contribution

It provides a necessary and sufficient condition for scattering in hyperbolic systems with integrable time-dependent coefficients, advancing understanding of their asymptotic behavior.

Findings

Solutions are asymptotically free if coefficients are Riemann integrable at infinity.

Nontrivial solutions are never asymptotically free if coefficients are not R-stable.

The scattering operator can be explicitly constructed based on the conditions.

Abstract

The present paper is devoted to finding a necessary and sufficient condition on the occurence of scattering for the regularly hyperbolic systems with time-dependent coefficients whose time-derivatives are integrable over the real line. More precisely, it will be shown that the solutions are asymptotically free if the coefficients are stable in the sense of the Riemann integrability as time goes to infinity, while each nontrivial solution is never asymptotically free provided that the coefficients are not R-stable as times goes to infinity. As a by-product, the scattering operator can be constructed. It is expected that the results obtained in the present paper would be brought into the study of the asymptotic behaviour of Kirchhoff systems.