  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.
