Searchlight asymptotics for high-frequency scattering by boundary inflection

The paper is devoted to an inner problem for a whispering gallery high-frequency asymptotic mode's scattering by a boundary inflection. The related boundary-value problem for a Schrödinger 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. The solution past the inflection point is shown to have a “searchlight” asymptotics corresponding to a beam concentrated near the limit ray. Certain decay and smoothness properties of the related searchlight amplitude are established. Further interpretations of the above result are also discussed: the existence of the associated generalised wave operator, and of a version of a unitary scattering operator connecting the modal and scattered asymptotic regimes.