The 2-dimensional hydrogen atom

In these notes we investigate a particular partial differential equation arising from the quantum mechanical description of a hydrogen atom. For simplicity we consider the atom in a 2-dimensional instead of a 3-dimensional universe. For convenience we put all fundamental constants of nature equal to 1. The problem. In quantum mechanics the electron is described by a so called wave function. A wave function is a complex valued function Ψ on R×R2 that gives the probability to find the electron at a certain position. To be more precise, suppose the position x0 of the electron is measured at time t0 and assume that U is a compact subset of R. Then the probability that the position x0 of the electron is inside U is given by