Limitations on the control of Schrödinger equations

Abstract

Abstract.We give the definitions of exact and approximate controllability for linear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical "No-go" result for evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non-controllability for the linear Schrödinger equation and distributed additive control, and we show that the Hartree equation of quantum chemistry with bilinear control (E(t) · x)u is not controllable in finite or infinite time. Finally, in Section 3, we give criteria for additive controllability of linear Schrödinger equations, and we give a distributed additive controllability result for the nonlinear Schrödinger equation if the data are small.Mathematics Subject Classification. 35Q40, 35Q55, 81Q99, 93B05.

Controllablity of a quantum particle in a 1D variable domain

“…Finally, non controllability results are proved in [37] and [28] for some particular linear and non linear Schrödinger equations. The result of [37] is discussed in Section 1.4.…”

Approximate Controllability, Exact Controllability, and Conical Eigenvalue Intersections for Quantum Mechanical Systems

“…See [17] and also [1,9,27,33,47] for further developments and related models. In the infinite-dimensional case, if the controlled Hamiltonians H 1 ,…,H m are bounded, exact controllability can be ruled out by functional analysis arguments [3,30,44]. Sufficient conditions for approximate controllability have been obtained by proving exact controllability of restrictions of (1) to spaces where the controlled Hamiltonians are unbounded [5][6][7].…”

Exact boundary controllability of the nonlinear Schrödinger equation

“…the Korteweg-de Vries equation (see [7,22,24,25,29,31]), or the Ginzburg-Landau equation (see [6,26]). Recently, Illner, Lange and Teismann [8,9] considered internal controllability of the nonlinear Schrödinger equation posed on a finite interval (−π , π): iv t + v xx + λ|v| 2 v = f (x, t), x ∈ (−π , π), (1.4) with the periodic boundary conditions v(−π , t) = v(π , t), v x (−π , t) = v x (π , t), (1.5) where the forcing function f = f (x, t), supported in a subinterval of (−π , π), is considered as a control input. They showed that the system (1.4)-(1.5) is locally exactly controllable in the space H 1 p (−π , π) := {v ∈ H 1 (−π , π): v(−π ) = v(π )}.…”