Oscillatory integral operators related to pointwise convergence of Schrödinger operators

[摘要] In this thesis we consider smooth analogues of operators studied in connectionwith the pointwise convergence of the solution, u(x,t), (x,t) ∈ ℝ^n x ℝ, of thefree Schrodinger equation to the given initial data. Such operators are interestingexamples of oscillatory integral operators with degenerate phase functions, and wedevelop 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-Sobolevspace H^8 (ℝ^n), where s depends on conditions on t.

[发布机构] University:California Institute of Technology;Department:Physics, Mathematics and Astronomy

[关键词] Mathematics