A remark on the attainable set of the Schrödinger equation

Jonas Lampart

doi:10.3934/eect.2020075

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2021, Volume 10, Issue 3: 461-469. Doi: 10.3934/eect.2020075

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A remark on the attainable set of the Schrödinger equation

Jonas Lampart

CNRS & Laboratoire interdisciplinaire Carnot de Bourgogne, UMR 6303 Université de Bourgogne Franche-Comté, 9 Av. A. Savary, 21078 Dijon Cedex, France

Received: October 31, 2019

Revised: March 31, 2020

Early access: June 2020

Published: September 2021

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Abstract

We discuss the set of wavefunctions $ \psi_V(t) $ that can be obtained from a given initial condition $ \psi_0 $ by applying the flow of the Schrödinger operator $ -\Delta + V(t,x) $ and varying the potential $ V(t,x) $. We show that this set has empty interior, both as a subset of the sphere in $ L^2( \mathbb{R}^d) $ and as a set of trajectories.

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Mathematics Subject Classification: Primary: 35Q93, 35Q41.

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