Energy Estimates for Low Regularity Bilinear Schr\"odinger Equations

TL;DR

This paper develops energy estimates for low regularity controls in bilinear quantum systems, enabling rigorous analysis and approximation of infinite-dimensional Schrödinger equations with unbounded potentials.

Contribution

It introduces new energy estimates based on total variation, facilitating the construction of propagators and error bounds for finite-dimensional approximations.

Findings

Energy estimates in terms of total variation of controls.

Rigorous construction of propagators for low regularity controls.

Error bounds for Galerkin finite-dimensional approximations.

Abstract

This paper presents an energy estimate in terms of the total variation of the control for bilinear infinite dimensional quantum systems with unbounded potentials. These estimates allow a rigorous construction of propagators associated with controls of bounded variation. Moreover, upper bounds of the error made when replacing the infinite dimensional system by its finite dimensional Galerkin approximations is presented.