Limitations on the control of Schrödinger equations

¹ Department of Mathematics and Statistics, University of Victoria, PO Box 3045, Victoria, B.C., V8W 3P4 Canada; rillner@math.uvic.ca
² Mathematisches Institut, Universität Köln, Weyertal 86-90, 50931 Köln, Germany; lange@mathematik.uni-koeln.de
³ Department of Mathematics and Statistics, Acadia University, Wolfville, N.S., B4P 1R6 Canada; hteisman@acadiau.ca

Received: 30 September 2004
Revised: 25 March 2005
Revised: 3 May 2005

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 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.