Generic controllability properties for the bilinear Schrödinger equation

In a recent paper we proposed a set of sufficient conditions for the approximate controllability of a discretespectrum bilinear Schrödinger equation on a fixed domain. These conditions are expressed in terms of the controlled potential and of the eigenpairs of the uncontrolled Schrödinger operator. The aim of this presentation is to show that these conditions are generic with respect to the uncontrolled or the controlled potential. The results are obtained by analytic perturbation arguments and through the study of asymptotic properties of eigenfunctions.

[1]  M. Reed.  Methods of Modern Mathematical Physics. I: Functional Analysis , 1972 .

[2]  Grégoire Allaire,et al.  Homogenization of periodic non self-adjoint problems with large drift and potential , 2007 .

[3]  Vahagn Nersesyan,et al.  Global approximate controllability for Schr\"odinger equation in higher Sobolev norms and applications , 2009, 0905.2438.

[4]  Robert Carlson.  Genericity of simple eigenvalues , 1979 .

[5]  Antoine Henrot,et al.  Variation et optimisation de formes : une analyse géométrique , 2005 .

[6]  M. Slemrod,et al.  Controllability of distributed bilinear systems , 1981, 1981 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes.

[7]  Mikhail Teytel,et al.  How rare are multiple eigenvalues , 1999 .

[8]  K. Friedrichs,et al.  Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren , 1934 .

[9]  Mario Sigalotti,et al.  Controllability of the discrete-spectrum Schrödinger equation driven by an external field , 2008, 0801.4893.

[10]  Claude Le Bris,et al.  Mathematical models and methods for ab initio quantum chemistry , 2000 .

[11]  Antoine Henrot,et al.  Extremum Problems for Eigenvalues of Elliptic Operators , 2006 .

[12]  Sylvain Ervedoza,et al.  Approximate Controllability for a System of Schrödinger Equations Modeling a Single Trapped Ion , 2009 .

[13]  Riccardo Adami,et al.  Controllability of the Schrödinger Equation via Intersection of Eigenvalues , 2005 .

[14]  Domenico D'Alessandro,et al.  Notions of controllability for bilinear multilevel quantum systems , 2003, IEEE Trans. Autom. Control..

[15]  U. Boscain,et al.  Controllability of the Schrödinger Equation via Intersection of Eigenvalues , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[16]  Karine Beauchard,et al.  Controllability of a quantum particle in a moving potential well , 2006 .

[17]  Gabriel Turinici,et al.  On the controllability of bilinear quantum systems , 2000 .

[18]  Roger W. Brockett,et al.  The Controllability of Infinite Quantum Systems and Closed Subspace Criteria , 2006 .

[19]  Manfred Requardt,et al.  PERTURBATION THEORY OF SCHRODINGER OPERATORS IN INFINITELY MANY COUPLING PARAMETERS , 1999 .

[20]  N. Khaneja,et al.  Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation , 2003, quant-ph/0302024.

[21]  C. Altafini,et al.  QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC) 2357 Controllability properties for finite dimensional quantum Markovian master equations , 2002, quant-ph/0211194.

[22]  Mario Sigalotti,et al.  The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent , 2008 .

[23]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[24]  Mazyar Mirrahimi,et al.  Lyapunov control of a quantum particle in a decaying potential , 2008, 0805.0910.

[25]  Karine Beauchard,et al.  Local controllability of a 1-D Schrödinger equation , 2005 .

[26]  S. S. Rodrigues.  Navier-Stokes Equation on the Rectangle: Controllability by Means of Low Mode Forcing , 2006 .

[27]  Andrey A. Agrachev,et al.  Controllability of 2D Euler and Navier-Stokes Equations by Degenerate Forcing , 2005 .

[28]  Marion Kee,et al.  Analysis , 2004, Machine Translation.

[29]  Antoine Henrot,et al.  Variation et optimisation de formes , 2005 .

[30]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[31]  J. H. Albert,et al.  Genericity of simple eigenvalues for elliptic PDE’s , 1975 .

[32]  B. Simon,et al.  Schrödinger Semigroups , 2007 .

[33]  Karine Beauchard,et al.  Spectral controllability for 2D and 3D linear Schrödinger equations , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[34]  K. Friedrichs,et al.  Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren , 1934 .

[35]  Mazyar Mirrahimi,et al.  Controllability of quantum harmonic oscillators , 2004, IEEE Transactions on Automatic Control.

[36]  Karen K. Uhlenbeck.  Generic Properties of Eigenfunctions , 1976 .

[37]  Shmuel Agmon,et al.  Bounds on exponential decay of eigenfunctions of Schrödinger operators , 1985 .

[38]  Mazyar Mirrahimi,et al.  Practical Stabilization of a Quantum Particle in a One-Dimensional Infinite Square Potential Well , 2009, SIAM J. Control. Optim..

[39]  Franz Rellich,et al.  Perturbation Theory of Eigenvalue Problems , 1969 .

[40]  Anna Maria Micheletti.  Metrica per famiglie di domini limitati e proprietà generiche degli autovalori , 1972 .

[41]  P. H. Müller,et al.  T. Kato, Perturbation theory for linear operators. (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Band 132) XX + 592 S. m. 3 Fig. Berlin/Heidelberg/New York Springer-Verlag. Preis geb. DM 79,20 , 1967 .

[42]  P. Anselone,et al.  Collectively Compact Operator Approximation Theory and Applications to Integral Equations , 1971 .

[43]  Anna Maria Micheletti.  Perturbazione dello spettro dell'operatore di Laplace, in relazione ad una variazione del campo , 1972 .

[44]  V. Nersesyan.  Growth of Sobolev Norms and Controllability of the Schrödinger Equation , 2008, 0804.3982.