Oscillatory integral operators related to pointwise convergence of Schrödinger operators

Citation

Kolasa, Lawrence A. (1994) Oscillatory integral operators related to pointwise convergence of Schrödinger operators. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/3j2h-q370. https://resolver.caltech.edu/CaltechTHESIS:05082013-093110717

Abstract

In this thesis we consider smooth analogues of operators studied in connection with the pointwise convergence of the solution, u(x,t), (x,t) ∈ ℝ^n x ℝ, of the free Schrodinger equation to the given initial data. Such operators are interesting examples of oscillatory integral operators with degenerate phase functions, and we develop strategies to capture the oscillations and obtain sharp L^2 → L^2 bounds. We then consider, for fixed smooth t(x), the restriction of u to the surface (x,t(x)). We find that u(x,t(x)) ∈ L^2(D^n) when the initial data is in a suitable L^2-Sobolev space H^8 (ℝ^n), where s depends on conditions on t.

Thesis Files