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Lecture notes from 2008 CMI/ETH Summer School on Evolution Equations. These notes are an informal introduction to the applications of microlocal methods in the study of linear evolution equations and spectral theory. Calculi of pseudodifferential operators and Fourier integral operators are discussed and axiomatized, but not constructed: the focus is on how to apply these tools.
This is a introductory course focusing some basic notions in pseudodifferential operators ($Psi$DOs) and microlocal analysis. We start this lecture notes with some notations and necessary preliminaries. Then the notion of symbols and $Psi$DOs are int
In this article, we consider a generalized Radon transform that comes up in ultrasound reflection tomography. In our model, the ultrasound emitter and receiver move at a constant distance apart along a circle. We analyze the microlocal properties of
We develop a second-microlocal calculus of pseudodifferential operators in the semiclassical setting. These operators test for Lagrangian regularity of semiclassical families of distributions on a manifold $X$ with respect to a Lagrangian submanifold
An analysis of the stability of the spindle transform, introduced in (Three dimensional Compton scattering tomography arXiv:1704.03378 [math.FA]), is presented. We do this via a microlocal approach and show that the normal operator for the spindle tr
We study initial boundary value problems for linear evolution partial differential equations (PDEs) posed on a time-dependent interval $l_1(t)<x<l_2(t)$, $0<t<T$, where $l_1(t)$ and $l_2(t)$ are given, real, differentiable functions, and $T$ is an ar