Ebru Toprak (UIUC and MSRI)

The next Analysis and PDE seminar will take place Monday, October 15, from 4-5pm in 740 Evans.

Title: The effect of threshold energy obstructions on the $L^1 \to L^\infty$
dispersive estimates for some Schrödinger type equations

Abstract: In this talk, I will discuss the differential equation $iu_t
= Hu, H := H_0 + V$ , where $V$ is a decaying potential and $H_0$ is a
Laplacian related operator. In particular, I will focus on when $H_0$
is Laplacian, Bilaplacian and Dirac operators. I will discuss how the
threshold energy obstructions, eigenvalues and resonances, effect the
$L^1 \to L^\infty$ behavior of $e^{itH} P_{ac} (H)$. The threshold
obstructions are known as the distributional solutions of $H\psi = 0$
in certain dimension dependent spaces. Due to its unwanted effects on
the dispersive estimates, its absence have been assumed in many
work. I will mention our previous results on Dirac operator and recent
results on Bilaplacian operator under different assumptions on
threshold energy obstructions.