Georgia Institute of Technology College of Sciences

I will begin by presenting our recent results on the spherically symmetric Coulomb waves. Specifically, we study the evolution operator of H= -\Delta+q/|x| where q>0. Utilizing a distorted Fourier transform adapted to H, we explicitly compute the evolution kernel. A detailed analysis of this kernel reveals that e^itH satisfies an L^1 \to L^{\infty} dispersive estimate with the natural decay rate t^{-3/2}. This work was conducted in collaboration with Adam Black, Bruno Vergara, and Jiahua Zhou. Following this, I will discuss our ongoing research on the nonlinear Schrödinger equation, where we apply the distorted Fourier transform developed for the Coulomb Hamiltonian. This work is being carried out in collaboration with Mengyi Xie.