Convergence of the time-discretized monotonic schemes

¹ Université Pierre et Marie Curie, Paris 6, Laboratoire Jacques-Louis Lions, 175 rue du Chevaleret 75013 Paris, France.
² Université Paris-Dauphine, Paris 9, CEREMADE, Place du Maréchal Lattre de Tassigny, 75775 Paris Cedex 16, France. Julien.Salomon@dauphine.fr

Received: 2 January 2006
Revised: 3 November 2006

Abstract

Many numerical simulations in (bilinear) quantum control use the monotonically convergent Krotov algorithms (introduced by Tannor et al. [Time Dependent Quantum Molecular Dynamics (1992) 347–360]), Zhu and Rabitz [J. Chem. Phys. (1998) 385–391] or their unified form described in Maday and Turinici [J. Chem. Phys. (2003) 8191–8196]. In Maday et al. [Num. Math. (2006) 323–338], a time discretization which preserves the property of monotonicity has been presented. This paper introduces a proof of the convergence of these schemes and some results regarding their rate of convergence.