# Control of the Schrödinger equation by slow deformations of the domain

### Alessandro Duca

Université de Lorraine, CNRS, Nancy, France### Romain Joly

Université Grenoble Alpes, CNRS, Institut Fourier, Grenoble, France### Dmitry Turaev

Imperial College London, UK; Higher School of Economics - Nizhny Novgorod, Russia

## Abstract

The aim of this work is to study the controllability of the Schrödinger equation $i\partial_t u(t)=-\Delta u(t)$ on $\Omega(t)$ with Dirichlet boundary conditions, where $\Omega(t)\subset \mathbb{R}^N$ is a time-varying domain. We prove the global approximate controllability of the equation in $L^2(\Omega)$, via an adiabatic deformation $\Omega(t)\subset\mathbb{R}^N$ ($t\in[0,T]$) such that $\Omega(0)=\Omega(T)=\Omega$. This control is strongly based on the Hamiltonian structure of the equation provided byby Duca and Joly [Ann. Henri Poincaré 22 (2021), 2029–2063], which enables the use of adiabatic motions.We also discuss several explicit interesting controls that we perform in the specific framework of rectangular domains.

## Cite this article

Alessandro Duca, Romain Joly, Dmitry Turaev, Control of the Schrödinger equation by slow deformations of the domain. Ann. Inst. H. Poincaré Anal. Non Linéaire (2023),

DOI 10.4171/AIHPC/86