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 boundary inflection. related boundary-value Schrödinger equation on half-line with potential linear in both space and time turns out be fundamental describing transitions from modal scattered patterns, despite having been intensively studied over several decades remains largely unsolved. solution past the inflection point shown have “searchlight” asymptotics corresponding beam concentrated near limit ray. Certain decay smoothness properties of searchlight amplitude are established. Further interpretations above result also discussed: existence associated generalized wave operator, version unitary operator connecting regimes.