Geometric Scattering Theory, Resolvent Estimates, and Wave Asymptotics

University Of Dayton, Dayton OH

Investigators

Abstract

This research project studies the propagation of waves in non-smooth media. Wave and Schrödinger equations are important partial differential equations that describe how waves, including sound, light, and gravitational waves, propagate in the natural world. Much is known about the behavior of solutions to these equations when the underlying media are smooth. However, most practical applications involve waves propagating in media that contain obstructions or discontinuities. The aim of this project is to resolve open questions about how waves scatter and decay when the underlying medium is not smooth. The results of the project are expected to inform predictions in several applications involving wave behavior in non-smooth media, including the propagation of seismic waves, the behavior of plasma, and light propagation in optical fibers. The project will contribute to the training of undergraduate and graduate students in a central, active area of mathematical physics. This project concerns research in geometric scattering theory. The project has two objectives. The first is to prove upper bounds on the resolvent of the semiclassical Schrödinger operator in limited regularity. Such resolvent bounds are a precursor to local energy decay rates for rough wave equations. The second objective is to use the recently developed Fredholm method to establish the existence of scattering-type eigenfunctions for a wide class of nonlinear Helmholtz equations. The main tools for this project come from semiclassical and microlocal analysis. In particular, the principal investigator aims to extend positive commutator and Carleman estimates, separation of variables, and b-vector field analysis to Schrödinger operators with low regularity coefficients. This will yield high frequency resolvent estimates, and in turn precise energy decay rates for waves traveling in heterogeneous media. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.