Valery Smyshlyaev (University College London)

An open problem in high-frequency diffraction: whispering gallery modes near boundary inflection

Abstract:
I will discuss a long open `canonical' problem in the high-frequency diffraction: a whispering gallery mode approaching a point of inflection of the boundary. It reduces to a Schoedinger equation in a half-plane $x>0$ with potential $xt$ and a prescribed asymptotics for $t\to -\infty$ in terms of a scaled Airy function. Some partial results are known, but the ultimate goal of finding the asymptotic for $t\to +\infty$ remains a challenge and may need some new ideas. I will try to provoke discussion of related spectral, scattering etc issues.