Control of the Schrödinger equation by slow deformations of the domain

The aim of this work is to study the controllability of the Schrödinger equation i @t u(t) D -Åu(t) on Ω(t) with Dirichlet boundary conditions, where Ω(t) ⊂ RN is a time-varying domain. We prove the global approximate controllability of the equation in L2(Ω), via an adiabatic deformation Ω(t) ⊂ RN (t 2 Œ0; T ç) such that Ω(0) D Ω(T) D Ω. This control is strongly based on the Hamiltonian structure of the equation provided by 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.