Schrödinger equations defined by a class of self-similar measures

Abstract

. We study linear and nonlinear Schr¨odinger equations defined by fractal measures. Under the assumption that the Laplacian has compact resolvent, we prove that there exists a uniqueness weak solution for a linear Schr¨odinger equation, and then use it to obtain the existence and uniqueness of weak solution of a nonlinear Schr¨odinger equation. We prove that for a class of self-similar measures on R with overlaps, the Schr¨odinger equations can be discretized so that the finite element method can be applied to obtain approximate solutions. We also prove that the numerical solutions converge to the actual solution and obtain the rate of convergence