Jared Wunsch, Northwestern University – Wunsch/PDE Mini-school

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Title: Trapping, diffraction, and decay of waves

Abstract: The long-time behavior of solutions to wave and Schrödinger equations is connected to geometry and dynamics via the correspondence principle, which states that at high-frequency, solutions propagate along classical particle orbits in phase space.  Making sense of the “high frequency” part of this statement often involves estimates for the resolvent operator family.  We will discuss some well-established results on how resolvent estimates and associated questions about distribution of scattering resonances are affected by classical dynamics, and then some recent results on what happens if the medium or manifold we are working on becomes singular, where suddenly the effect of diffraction come into play as a correction to the usual correspondence principle.

Lecture 1: Introduction to semiclassical analysis — eigenfunctions, semiclassical operators; WKB solutions; semiclassical pseudodifferential operators and their key properties (including functional calculus); applications to Weyl’s law.

Lecture 2: Hamilton dynamics; defect measures and semiclassical wavefront set; propagation of both

Lecture 3: Propagation continued; applications to nontrapping resolvent estimates on the real axis; damped wave equation; mild trapping

Lecture 4: Resonances, nontrapping / trapping, diffractive trapping