Wunsch Mini-school Schedule
Wednesday, Dec. 5 | Thursday, Dec. 6 | Friday, Dec. 7 | |
9:30-10:00 | Coffee and scones (Phillips 330) | Coffee and scones (Phillips 330) | Coffee and scones (Phillips 330) |
10:00-11:00 | Wunsch Lecture 1 (Phillips 383) | Wunsch Lecture 3 (Phillips 383) | Wunsch Lecture 4 (Phillips 383) |
11:00-12:00 | Problem session (Phillips 383) | ||
12:00-1:00 | |||
1:00-2:00 | Wunsch Lecture 2 (Phillips 383) | Problem session (Phillips 383) | |
2:00-3:00 | Problem session (Phillips 367) | ||
3:00-4:00 | Kleinhenz Lecture (Phillips 383) | ||
4:00-5:00 | Gannot Lecture (Phillips 383) | Galkowski Lecture (Phillips 383) |
Titles / Abstracts:
Jared Wunsch, Northwestern University
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
Oran Gannot, Northwestern University
Title: Scattering resonances generated by diffractive trapping
Abstract: In potential scattering where classical particles with a certain energy escape to infinity, it is expected that quantum states with neighboring energies decay rapidly in time. We will discuss some known results where the decay rate depends on the regularity of the potential. In cases where irregularities of the potential occur along an interface, the decay rate can be understood in terms of the strength of diffractive effects. We will illustrate this explicitly in one dimensional examples.
Perry Kleinhenz, Northwestern University
Title: Stabilization rates for the damped wave equation with Hölder regular damping
Abstract: We study the decay rate of the energy of solutions to the damped wave equation in a setup where the geometric control condition is violated. In particular we consider the case of a torus where the damping is $0$ on a strip and vanishes like a polynomial $x^{\beta}$. We prove that the semigroup is stable at rate at least as fast as $1/t^{(beta+2)/(\beta+4)}$ and sketch a proof that the semigroup decays no faster than $1/t^{(\beta+2)/(\beta+3)}$. These results establish an explicit relation between the rate of vanishing of the damping and rate of decay of solutions.
Jeffrey Galkowski, Northeastern University
Title: Optimal Resolvent Estimates in Non-trapping Geometries
Abstract: In this talk we discuss how propagation estimates can be used to prove resolvent estimates in non-trapping geometries. We will introduce the notion of defect measures and prove propagation estimates in this setting. We will then use these estimates to prove a non-trapping resolvent estimate and finally refine the estimates to give optimal bounds on the resolvent. Time permitting we discuss the necessary modification when the manifold has a boundary.