The Schrödinger Equaton and Uncertainty Principles

Abstract

The main task of this thesis is the analysis of the initial data u0 of Schrödinger’s initial value problem in order to determine certain properties of its dynamical evolution. First, we consider the elliptic Schrödinger problem in its perturbative form. The idea is to find lower bounds for the solution giving conditions at time t = 0 together with a size condition on the potential. After analyzing the elliptic case, we give a similar result for the hyperbolic Schrödinger operator. Next, we focus on the free particle case; this is the case where no potential is involved. The goal here is to quantify the L2 norm of the solution in a space-time cylinder. Following the same idea as before we want to find conditions at time t = 0 to ensure this. To carry out this task we define the Σδ space where δ is a parameter on the interval (0, 1]. We see that if belongs in this space then so does its evolution in time and use this fact to give lower bounds for the L2 norm of the solution. For δ = 1 we give a different approach and make use of the Virial Theorem. We will see that this case has particular properties. Finally, we study dynamical uncertainty principles derived from the previous study. The key point will be to write the solution as u = ρeiθ, where ρ and θ are real functions. Thus, we give uncertainty principles in terms of these functions and find explicit expressions for them so that u becomes a minimizer of the problem.