On the Schrӧdinger equation with singular potentials

Abstract

We study the Cauchy problem for the non-linear Schrӧdinger equation with singular potentials. For the point-mass potential and nonperiodic case, we prove existence and asymptotic stability of global solutions in weak-$L^{p}$ spaces. Specific interest is given to the point-like $\delta$ and $\delta'$ impurity and to two $\delta$-interactions in one dimension. We also consider the periodic case, which is analyzed in a functional space based on Fourier transform and local-in-time well-posedness is proved.