Searchlight Asymptotics for High-Frequency Scattering by Boundary Inflection

TL;DR

This paper analyzes high-frequency wave scattering near boundary inflections, revealing a 'searchlight' asymptotic pattern that concentrates near a limit ray, advancing understanding of modal-to-scattered wave transitions.

Contribution

It proves the existence of searchlight asymptotics for scattering by boundary inflections and explores related wave operators, addressing a longstanding open problem in wave scattering theory.

Findings

Solution exhibits searchlight asymptotics near the limit ray.

Decay and smoothness properties of the searchlight amplitude are established.

Discussion of generalized wave operators and scattering operators connecting regimes.

Abstract

We consider an inner problem for whispering gallery high-frequency asymptotic mode's scattering by a boundary inflection. The related boundary-value problem for a Schr\"{o}dinger equation on a half-line with a potential linear in both space and time appears fundamental for describing transitions from modal to scattered asymptotic patterns, and despite having been intensively studied over several decades remains largely unsolved. We prove that the solution past the inflection point has a ``searchlight'' asymptotics corresponding to a beam concentrated near the limit ray, and establish certain decay and smoothness properties of the related searchlight amplitude. We also discuss further interpretations of the above result: the existence of associated generalised wave operator, and of a version of a unitary scattering operator connecting the modal and scattered asymptotic regimes.