Computability of magnetic Schr\"odinger and Hartree equations on unbounded domains

TL;DR

This paper investigates the computability of solutions to magnetic Schr"odinger and Hartree equations on unbounded domains, establishing conditions for global computation with error control and uniform runtime, with applications in optimal control.

Contribution

It demonstrates that solutions can be globally computed with error bounds given decay estimates and introduces methods for uniform computational runtime for various initial states and potentials.

Findings

Solutions are computable with error control under decay estimates.

Uniform computational runtime is achievable for different initial states and potentials.

Applications in optimal control are explored with numerical examples.

Abstract

We study the computability of global solutions to linear Schr\"odinger equations with magnetic fields and the Hartree equation on $\mathbb R^3$. We show that the solution can always be globally computed with error control on the entire space if there exist a priori decay estimates in generalized Sobolev norms on the initial state. Using weighted Sobolev norm estimates, we show that the solution can be computed with uniform computational runtime with respect to initial states and potentials. We finally study applications in optimal control theory and provide numerical examples.