Space-time methods for Schrödinger equations

The time-dependent Schrödinger equation plays an important role in many areas of physics. It is the central governing equation in quantum mechanics. In other words, the solution to the time-dependent Schrödinger equation is of utmost importance. Unfortunately, the exact solution is not generally accessible. However, an approximation of this solution can be calculated using numerical methods. Established methods are based on separating the numerical methods into a temporal and a spatial part, i.e., treating space and time differently. This leads to numerical methods in which the approximation at a particular point in time depends only on points in the past. To exploit the capacity of modern supercomputers, methods are needed to calculate the temporal part in parallel. Time parallelization is difficult to achieve with the established methods, which are sequential concerning time. The situation is different with so-called space-time methods, which treat space and time equally. In particular, time is interpreted as an additional part of space. In this project, space-time methods will be developed to approximate the time-dependent Schrödinger equation. First, the unique solvability of the time-dependent Schrödinger equation will be investigated. The aim will be to derive a new characterization of the solutions of the time-dependent Schrödinger equation. Based on this, the focus will lay on the development of different space-time methods. The project will use polynomials in space and time to approximate the solution of the time-dependent Schrödinger equation. A rigorous analysis of these methods will play a major role and will be the aim of this project. In particular, the unique solvability of the proposed methods will be examined in detail. Furthermore, the error caused by approximating the solution will be investigated. In addition, the time parallelization of these methods will be considered. The developed space-time methods will be implemented and extensively tested.