An Ultra-Weak Space-Time Variational Formulation for the Schr\"odinger Equation

TL;DR

This paper introduces a novel ultra-weak space-time variational formulation for the time-dependent Schrödinger equation, demonstrating stability, norm-preservation, and effective numerical performance, especially with non-smooth initial data.

Contribution

It develops a new ultra-weak variational formulation with optimal stability and norm-preservation properties for the Schrödinger equation, including discretization methods and numerical validation.

Findings

Optimal inf-sup stability of the formulation.

Norm-preservation in the ultra-weak and discretized schemes.

Effective numerical performance with non-smooth initial data.

Abstract

We present a well-posed ultra-weak space-time variational formulation for the time-dependent version of the linear Schr\"odinger equation with an instationary Hamiltonian. We prove optimal inf-sup stability and introduce a space-time Petrov-Galerkin discretization with optimal discrete inf-sup stability. We show norm-preservation of the ultra-weak formulation. The inf-sup optimal Petrov-Galerkin discretization is shown to be asymptotically norm-preserving, where the deviation is shown to be in the order of the discretization. In addition, we introduce a Galerkin discretization, which has suboptimal inf-sup stability but exact norm-preservation. Numerical experiments underline the performance of the ultra-weak space-time variational formulation, especially for non-smooth initial data.