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

Seminar | October 15 | 4:10-5 p.m. | 740 Evans Hall

 Ebru Toprak, UIUC and MSRI

 Department of Mathematics

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.