Spectral controllability for 2D and 3D linear Schrödinger equations

Abstract

We consider a quantum particle in an infinite square potential well of R n , n = 2 , 3 , subjected to a control which is a uniform (in space) electric field. Under the dipolar moment approximation, the wave function solves a PDE of Schrödinger type. We study the spectral controllability in finite time of the linearized system around the ground state. We characterize one necessary condition for spectral controllability in finite time: ( Kal) if Ω is the bottom of the well, then for every eigenvalue λ of − Δ Ω D , the projections of the dipolar moment onto every (normalized) eigenvector associated to λ are linearly independent in R n . In 3D, our main result states that spectral controllability in finite time never holds for one-directional dipolar moment. The proof uses classical results from trigonometric moment theory and properties about the set of zeros of entire functions. In 2D, we first prove the existence of a minimal time T min ( Ω ) > 0 for spectral controllability, i.e., if T > T min ( Ω ) , one has spectral controllability in time T if condition ( Kal) holds true for ( Ω) and, if T < T min ( Ω ) and the dipolar moment is one-directional, then one does not have spectral controllability in time T. We next characterize a necessary and sufficient condition on the dipolar moment insuring that spectral controllability in time T > T min ( Ω ) holds generically with respect to the domain. The proof relies on shape differentiation and a careful study of Dirichlet-to-Neumann operators associated to certain Helmholtz equations. We also show that one can recover exact controllability in abstract spaces from this 2D spectral controllability, by adapting a classical variational argument from control theory.