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Controllability of the discrete-spectrum Schrödinger equation driven by an external field. (English) Zbl 1161.35049
Summary: We prove approximate controllability of the bilinear Schrödinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the Galerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential, both controlled by suitable potentials.

##### MSC:
 35Q55 NLS equations (nonlinear Schrödinger equations) 93C20 Control/observation systems governed by partial differential equations 93B05 Controllability 93B40 Computational methods in systems theory (MSC2010)
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##### References:
 [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] 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, 409-441, (2006) · Zbl 1106.93006 [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, 399-415, (2007) · Zbl 1177.60059 [4] Agrachev, A.A.; Sachkov, Y.L., Control theory from the geometric viewpoint, Encyclopaedia of mathematical sciences, vol. 87, (2004), Springer-Verlag Berlin, Control Theory and Optimization, II · Zbl 1062.93001 [5] Agrachev, A.A.; Sarychev, A.V., Controllability of 2D Euler and navier – stokes equations by degenerate forcing, Commun. math. phys., 265, 3, 673-697, (2006) · Zbl 1105.93008 [6] Albert, J.H., Genericity of simple eigenvalues for elliptic PDE’s, Proc. amer. math. soc., 48, 413-418, (1975) · Zbl 0302.35071 [7] Albertini, F.; D’Alessandro, D., Notions of controllability for bilinear multilevel quantum systems, IEEE trans. automat. control, 48, 8, 1399-1403, (2003) · Zbl 1364.93059 [8] Altafini, C., Controllability of quantum mechanical systems by root space decomposition of $$\mathit{su}(N)$$, J. math. phys., 43, 5, 2051-2062, (2002) · Zbl 1059.93016 [9] Altafini, C., Controllability properties for finite dimensional quantum Markovian master equations, J. math. phys., 44, 6, 2357-2372, (2003) · Zbl 1062.82033 [10] Ball, J.M.; Marsden, J.E.; Slemrod, M., Controllability for distributed bilinear systems, SIAM J. control optim., 20, 4, 575-597, (1982) · Zbl 0485.93015 [11] Baudouin, L.; Kavian, O.; Puel, J.-P., Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control, J. differential equations, 216, 1, 188-222, (2005) · Zbl 1109.35094 [12] Beauchard, K., Local controllability of a 1-D Schrödinger equation, J. math. pures appl. (9), 84, 7, 851-956, (2005) · Zbl 1124.93009 [13] Beauchard, K.; Coron, J.-M., Controllability of a quantum particle in a moving potential well, J. funct. anal., 232, 2, 328-389, (2006) · Zbl 1188.93017 [14] Borzì, A.; Decker, E., Analysis of a leap-frog pseudospectral scheme for the Schrödinger equation, J. comput. appl. math., 193, 1, 65-88, (2006) · Zbl 1118.65107 [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. B, 5, 4, 957-990, (2005), (electronic) · Zbl 1084.81083 [16] Boscain, U.; Charlot, G., Resonance of minimizers for n-level quantum systems with an arbitrary cost, ESAIM control optim. calc. var., 10, 4, 593-614, (2004), (electronic) · Zbl 1072.49002 [17] Boscain, U.; Mason, P., Time minimal trajectories for a spin 1/2 particle in a magnetic field, J. math. phys., 47, 6, 29, (2006), 062101 · Zbl 1112.81098 [18] Coron, J.-M., Control and nonlinearity, Mathematical surveys and monographs, vol. 136, (2007), American Mathematical Society Providence, RI [19] D’Alessandro, D., Introduction to quantum control and dynamics, Applied mathematics and nonlinear science series, (2008), Chapman, Hall/CRC Boca Raton, FL · Zbl 1139.81001 [20] Davies, E.B., Spectral theory and differential operators, Cambridge studies in advanced mathematics, vol. 42, (1995), Cambridge University Press Cambridge · Zbl 0893.47004 [21] Henrot, A., Extremum problems for eigenvalues of elliptic operators, Frontiers in mathematics, (2006), Birkhäuser Verlag Basel · Zbl 1109.35081 [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, 2056-2059, (2000) [23] Ito, K.; Kunisch, K., Optimal bilinear control of an abstract Schrödinger equation, SIAM J. control optim., 46, 1, 274-287, (2007), (electronic) · Zbl 1136.35089 [24] Jurdjevic, V.; Sussmann, H.J., Control systems on Lie groups, J. differential equations, 12, 313-329, (1972) · Zbl 0237.93027 [25] Kato, T., Perturbation theory for linear operators, Die grundlehren der mathematischen wissenschaften, Band 132, (1966), Springer-Verlag New York · Zbl 0148.12601 [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. A, 65, 3, 11, (2002), 032301 [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] Mirrahimi, M.; Rouchon, P., Controllability of quantum harmonic oscillators, IEEE trans. automat. control, 49, 5, 745-747, (2004) · Zbl 1365.81065 [29] Peirce, A.; Dahleh, M.; Rabitz, H., Optimal control of quantum mechanical systems: existence, numerical approximations, and applications, Phys. rev. A, 37, 4950-4964, (1988) [30] Pierfelice, V., Strichartz estimates for the Schrödinger and heat equations perturbed with singular and time dependent potentials, Asymptotic anal., 47, 1-2, 1-18, (2006) · Zbl 1100.35020 [31] Rabitz, H.; de Vivie-Riedle, H.; Motzkus, R.; Kompa, K., Wither the future of controlling quantum phenomena?, Science, 288, 824-828, (2000) [32] Reed, M.; Simon, B., Methods of modern mathematical physics. IV. analysis of operators, (1978), Academic Press (Harcourt Brace Jovanovich Publishers) New York · Zbl 0401.47001 [33] Rellich, F., Perturbation theory of eigenvalue problems, (1969), Gordon Breach Science Publishers New York, Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz · Zbl 0181.42002 [34] Rodnianski, I.; Schlag, W., Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. math., 155, 3, 451-513, (2004) · Zbl 1063.35035 [35] Rodrigues, S.S., Navier – stokes equation on the rectangle controllability by means of low mode forcing, J. dynam. control syst., 12, 4, 517-562, (2006) · Zbl 1105.35085 [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 [37] Sachkov, Y.L., Controllability of invariant systems on Lie groups and homogeneous spaces, J. math. sci. (New York), 100, 4, 2355-2427, (2000), Dynamical systems, 8 · Zbl 1073.93511 [38] Shapiro, M.; Brumer, P., Principles of the quantum control of molecular processes, (2003), Wiley-VCH, pp. 250 [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 · Zbl 1140.93395 [40] Turinici, G., On the controllability of bilinear quantum systems, () [41] Zuazua, E., Remarks on the controllability of the Schrödinger equation, (), 193-211
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