Schrödinger equations defined by a class of self-similar measures

We study linear and non-linear Schrödinger equations defined by fractal measures. Under the assumption that the Laplacian has compact resolvent, we prove that there exists a unique weak solution for a linear Schrödinger equation, and then use it to obtain the existence and uniqueness of a weak solution of a non-linear Schrödinger equation. We prove that for a class of self-similar measures on R with overlaps, the linear Schrödinger 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.