Controllability of the Schrödinger Equation via Intersection of Eigenvalues

We introduce two models of controlled infinite dimensional quantum system whose Hamiltonian operator has a purely discrete spectrum. For any couple of eigenstates we construct a path in the space of controls that approximately steers the system from one eigenstate to the other. To this purpose we use the adiabatic theory for quantum systems, and therefore the strategy requires large times. keywords: Quantum Control, Controllability of PDEs, Adiabatic Theory, δ-like Interactions

[1]  Stefan Teufel,et al.  Adiabatic perturbation theory in quantum dynamics , 2003 .

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

[3]  Ionization for Three Dimensional Time-Dependent Point Interactions , 2004, math-ph/0402011.

[4]  Ruedi Seiler,et al.  Adiabatic theorems and applications to the quantum hall effect , 1987 .

[5]  Tosio Kato.  On the Adiabatic Theorem of Quantum Mechanics , 1950 .

[6]  Domenico D'Alessandro,et al.  Optimal control of two-level quantum systems , 2001, IEEE Trans. Autom. Control..

[7]  H. Jauslin,et al.  Control of Quantum Dynamics by Laser Pulses: Adiabatic Floquet Theory , 2003 .

[8]  Evolution of a Model Quantum System¶ Under Time Periodic Forcing:¶Conditions for Complete Ionization , 2000, math-ph/0011001.

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

[10]  Bruce W. Shore,et al.  Theory of Coherent Atomic Excitation , 1991 .

[11]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[12]  R. Brockett,et al.  Time optimal control in spin systems , 2000, quant-ph/0006114.

[13]  Pierre Rouchon.  Control of a Quantum Particle in a Moving Potential Well , 2003 .

[14]  George A. Hagedorn,et al.  Molecular propagation through electron energy level crossings , 1994 .

[15]  Ugo Boscain,et al.  Resonance of minimizers for n-level quantum systems with an arbitrary cost , 2003 .

[16]  Sergio Albeverio,et al.  Singular perturbations of differential operators : solvable Schrödinger type operators , 2000 .

[17]  L. P. Yatsenko,et al.  Topology of adiabatic passage , 2002 .

[18]  H. Holden,et al.  Solvable Models in Quantum Mechanics: Second Edition , 2004 .

[19]  R. Adami.  A Class of Nonlinear Schro dinger Equations with Concentrated Nonlinearity , 2001 .

[20]  Sergio Albeverio,et al.  Singular Perturbations of Differential Operators , 2000 .

[21]  Sergio Albeverio,et al.  Solvable Models in Quantum Mechanics , 1988 .

[22]  U. Gaubatz,et al.  Population switching between vibrational levels in molecular beams , 1988 .

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