Controllability of the discrete-spectrum Schrödinger equation driven by an external field

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1. [1] R. Adami, U. Boscain, Controllability of the Schrödinger equation via intersection of eigenvalues, in: Proceedings of the 44th IEEE Conference on Decision and Control, December 12–15, 2005, pp. 1080–1085. 
2. [2] Agrachev A., Chambrion T., An estimation of the controllability time for single-input systems on compact Lie groups, ESAIM Control Optim. Calc. Var.12 (3) (2006) 409-441. Zbl1106.93006MR2224821
3. [3] Agrachev A., Kuksin S., Sarychev A., Shirikyan A., On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier–Stokes equations, Ann. Inst. H. Poincaré Probab. Statist.43 (4) (2007) 399-415. Zbl1177.60059MR2329509
4. [4] Agrachev A.A., Sachkov Y.L., Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, vol. 87, Springer-Verlag, Berlin, 2004, Control Theory and Optimization, II. Zbl1062.93001MR2062547
5. [5] Agrachev A.A., Sarychev A.V., Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Commun. Math. Phys.265 (3) (2006) 673-697. Zbl1105.93008MR2231685
6. [6] Albert J.H., Genericity of simple eigenvalues for elliptic PDE's, Proc. Amer. Math. Soc.48 (1975) 413-418. Zbl0302.35071MR385934
7. [7] Albertini F., D'Alessandro D., Notions of controllability for bilinear multilevel quantum systems, IEEE Trans. Automat. Control48 (8) (2003) 1399-1403. MR2004373
8. [8] Altafini C., Controllability of quantum mechanical systems by root space decomposition of $\mathrm{su}\left(N\right)$, J. Math. Phys.43 (5) (2002) 2051-2062. Zbl1059.93016MR1893660
9. [9] Altafini C., Controllability properties for finite dimensional quantum Markovian master equations, J. Math. Phys.44 (6) (2003) 2357-2372. Zbl1062.82033MR1979090
10. [10] Ball J.M., Marsden J.E., Slemrod M., Controllability for distributed bilinear systems, SIAM J. Control Optim.20 (4) (1982) 575-597. Zbl0485.93015MR661034
11. [11] Baudouin L., Kavian O., Puel J.-P., Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control, J. Differential Equations216 (1) (2005) 188-222. Zbl1109.35094MR2158922
12. [12] Beauchard K., Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl. (9)84 (7) (2005) 851-956. Zbl1124.93009MR2144647
13. [13] Beauchard K., Coron J.-M., Controllability of a quantum particle in a moving potential well, J. Funct. Anal.232 (2) (2006) 328-389. Zbl1188.93017MR2200740
14. [14] Borzì A., Decker E., Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation, J. Comput. Appl. Math.193 (1) (2006) 65-88. Zbl1118.65107MR2228707
15. [15] Boscain U., Chambrion T., Charlot G., Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy, Discrete Contin. Dyn. Syst. Ser. B5 (4) (2005) 957-990, (electronic). Zbl1084.81083MR2170218
16. [16] Boscain U., Charlot G., Resonance of minimizers for n-level quantum systems with an arbitrary cost, ESAIM Control Optim. Calc. Var.10 (4) (2004) 593-614, (electronic). Zbl1072.49002MR2111082
17. [17] Boscain U., Mason P., Time minimal trajectories for a spin 1/2 particle in a magnetic field, J. Math. Phys.47 (6) (2006) 29, 062101. Zbl1112.81098MR2239948
18. [18] Coron J.-M., Control and Nonlinearity, Mathematical Surveys and Monographs, vol. 136, American Mathematical Society, Providence, RI, 2007. Zbl1140.93002MR2302744
19. [19] D'Alessandro D., Introduction to Quantum Control and Dynamics, Applied Mathematics and Nonlinear Science Series, Chapman, Hall/CRC, Boca Raton, FL, 2008. Zbl1139.81001MR2357229
20. [20] Davies E.B., Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. Zbl0893.47004MR1349825
21. [21] Henrot A., Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2006. Zbl1109.35081MR2251558
22. [22] Hübler P., Bargon J., Glaser S.J., Nuclear magnetic resonance quantum computing exploiting the pure spin state of para hydrogen, J. Chem. Phys.113 (6) (2000) 2056-2059. 
23. [23] Ito K., Kunisch K., Optimal bilinear control of an abstract Schrödinger equation, SIAM J. Control Optim.46 (1) (2007) 274-287, (electronic). Zbl1136.35089MR2299629
24. [24] Jurdjevic V., Sussmann H.J., Control systems on Lie groups, J. Differential Equations12 (1972) 313-329. Zbl0237.93027MR331185
25. [25] Kato T., Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag, New York, 1966. Zbl0148.12601MR203473
26. [26] Khaneja N., Glaser S.J., Brockett R., Sub-Riemannian geometry and time optimal control of three spin systems: quantum gates and coherence transfer, Phys. Rev. A65 (3) (2002) 11, 032301. MR1891763
27. [27] M. Mirrahimi, Lyapunov control of a particle in a finite quantum potential well, in: Proceedings of the 45th IEEE Conference on Decision and Control, December 13–15, 2006. 
28. [28] Mirrahimi M., Rouchon P., Controllability of quantum harmonic oscillators, IEEE Trans. Automat. Control49 (5) (2004) 745-747. MR2057808
29. [29] Peirce A., Dahleh M., Rabitz H., Optimal control of quantum mechanical systems: Existence, numerical approximations, and applications, Phys. Rev. A37 (1988) 4950-4964. MR949169
30. [30] Pierfelice V., Strichartz estimates for the Schrödinger and heat equations perturbed with singular and time dependent potentials, Asymptotic Anal.47 (1–2) (2006) 1-18. Zbl1100.35020MR2224403
31. [31] Rabitz H., de Vivie-Riedle H., Motzkus R., Kompa K., Wither the future of controlling quantum phenomena?, Science288 (2000) 824-828. 
32. [32] Reed M., Simon B., Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press (Harcourt Brace Jovanovich Publishers), New York, 1978. Zbl0401.47001MR493421
33. [33] Rellich F., Perturbation Theory of Eigenvalue Problems, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon Breach Science Publishers, New York, 1969. Zbl0181.42002MR240668
34. [34] Rodnianski I., Schlag W., Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math.155 (3) (2004) 451-513. Zbl1063.35035MR2038194
35. [35] Rodrigues S.S., Navier–Stokes equation on the rectangle controllability by means of low mode forcing, J. Dynam. Control Syst.12 (4) (2006) 517-562. Zbl1105.35085MR2253360
36. [36] P. Rouchon, Control of a quantum particle in a moving potential well, in: Lagrangian and Hamiltonian Methods for Nonlinear Control 2003, IFAC, Laxenburg, 2003, pp. 287–290. MR2082989
37. [37] Sachkov Y.L., Controllability of invariant systems on Lie groups and homogeneous spaces, Dynamical systems, 8, J. Math. Sci. (New York)100 (4) (2000) 2355-2427. Zbl1073.93511MR1776551
38. [38] Shapiro M., Brumer P., Principles of the Quantum Control of Molecular Processes, Wiley-VCH, 2003, pp. 250. Zbl1247.81006
39. [39] G. Tenenbaum, M. Tucsnak, K. Ramdani, T. Takahashi, A spectral approach for the exact observability of infinite dimensional systems with skew-adjoint generator, J. Funct. Anal., 2007. Zbl1140.93395MR2158180
40. [40] Turinici G., On the controllability of bilinear quantum systems, in: Defranceschi M., Le Bris C. (Eds.), Mathematical Models and Methods for ab initio Quantum Chemistry, Lecture Notes in Chemistry, vol. 74, Springer, 2000. Zbl1007.81019MR1857459
41. [41] Zuazua E., Remarks on the controllability of the Schrödinger equation, in: Quantum Control: Mathematical and Numerical Challenges, CRM Proc. Lecture Notes, vol. 33, Amer. Math. Soc., Providence, RI, 2003, pp. 193-211. MR2043529

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