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Growth of Sobolev norms and controllability of the Schrödinger equation. (English) Zbl 1180.93017
Summary: We obtain a stabilization result for both linear and nonlinear Schrödinger equations under generic assumptions on the potential. Then we consider the Schrödinger equations with a potential which has a random time-dependent amplitude. We show that if the distribution of the amplitude is sufficiently non-degenerate, then any trajectory of the system is almost surely non-bounded in Sobolev spaces.

93B05 Controllability
93C20 Control/observation systems governed by partial differential equations
35Q40 PDEs in connection with quantum mechanics
93D21 Adaptive or robust stabilization
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[1] Agrachev A., Chambrion T.: An estimation of the controllability time for single-input systems on compact Lie groups. J. ESAIM Control Optim. Calc. Var. 12(3), 409–441 (2006) · Zbl 1106.93006
[2] Albert J.H.: Genericity of simple eigenvalues for elliptic PDE’s. Proc. Amer. Math. Soc. 48, 413–418 (1975) · Zbl 0302.35071
[3] Albertini F., D’Alessandro D.: Notions of controllability for bilinear multilevel quantum systems. IEEE Transactions on Automatic Control 48(8), 1399–1403 (2003) · Zbl 1364.93059
[4] Altafini C.: Controllability of quantum mechanical systems by root space decomposition of su(n). J. Math. Phys. 43(5), 2051–2062 (2002) · Zbl 1059.93016
[5] Ball J.M., Marsden J.E., Slemrod M.: Controllability for distributed bilinear systems. SIAM J. Control Optim. 20, 575–597 (1982) · Zbl 0485.93015
[6] Baudouin L., Puel J.-P.: Uniqueness and stability in an inverse problem for the Schrödinger equation. Inverse Problems 18, 1537–1554 (2001) · Zbl 1023.35091
[7] Beauchard K.: Local controllability of a 1-D Schrödinger equation. J. Math. Pures et Appl. 84(7), 851–956 (2005) · Zbl 1124.93009
[8] 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
[9] Beauchard K., Coron J.-M., Mirrahimi M., Rouchon P.: Implicit Lyapunov control of finite dimensional Schrödinger equations. Syst. Cont. Lett. 56, 388–395 (2007) · Zbl 1110.81063
[10] Beauchard, K., Mirrahimi, M.: Approximate stabilization of a quantum particle in a 1D infinite square potential well. http://arxiv.org/abs/0801.1522v1[math.AP] , 2008, to apppear SIAMJ Cont. Opt. · Zbl 1194.93176
[11] Bourgain J.: On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential. J. Anal. Math. 77, 315–348 (1999) · Zbl 0938.35026
[12] Burq N.: Contrôle de l’équation des plaques en présence d’obstacles strictement convexes. Mémoire de la S.M.F. 55, 126 (1993) · Zbl 0930.93007
[13] Cazenave, T.: Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, 10, Providence, RI: Amer. Math. Soc., 2003 · Zbl 1055.35003
[14] Chambrion, T., Mason, P., Sigalotti, M., Boscain, U.: Controllability of the discrete-spectrum Schrödinger equation driven by an external field. http://arxiv.org/abs/0801.4893v3[math.OC] , 2008 · Zbl 1161.35049
[15] Coron, J.-M.: Control and nonlinearity. Mathematical Surveys and Monographs, Providence, RI: Amer. Math. Soc., 136, 2007 · Zbl 1140.93002
[16] Dehman B., Gérard P., Lebeau G.: Stabilization and control for the nonlinear Schrödinger equation on a compact surface. Math. Z. 254(4), 729–749 (2006) · Zbl 1127.93015
[17] Eliasson L.H., Kuksin S.B.: On reducibility of Schrödinger equations with quasiperiodic in time potentials. Commun. Math. Phys. 286(1), 125–135 (2009) · Zbl 1176.35141
[18] Erdogan M.B., Killip R., Schlag W.: Energy growth in Schrödinger’s equation with Markovian forcing. Commun. Math. Phys. 240, 1–29 (2003) · Zbl 1033.81023
[19] Jerison D., Kenig C.E.: Unique continuation and absence of positive eigenvalues for Schrödinger operators (with an appendix by E. M. Stein). Ann. Math. 121(3), 463–494 (1985) · Zbl 0593.35119
[20] Lebeau G.: Contrôle de l’équation de Schrödinger. J. Math. Pures Appl. 71, 267–291 (1992) · Zbl 0838.35013
[21] Machtyngier E., Zuazua E.: Stabilization of the Schrödinger equation. Portugaliae Matematica 51(2), 243–256 (1994) · Zbl 0814.35008
[22] Mirrahimi, M.: Lyapunov control of a particle in a finite quantum potential well. IEEE Conf. on Decision and Control, San Diego, 2006
[23] Nersesyan, V.: Exponential mixing for finite-dimensional approximations of the Schrödinger equation with multiplicative noise. http://arxiv.org/abs/0710.3693v1[math-ph] , 2007 · Zbl 1178.37010
[24] Nersesyan, V.: Global approximate controllability for Schrödinger equation in higher Sobolev norms. In preparation, 2009 · Zbl 1180.93017
[25] Øksendal B.: Stochastic Differential Equations. Springer–Verlag, Berlin-Heidelberg-New York (2003)
[26] Pöschel J., Trubowitz E.: Inverse Spectral Theory. Academic Press, New York (1987) · Zbl 0623.34001
[27] Ramakrishna V., Salapaka M., Dahleh M., Rabitz H., Pierce A.: Controllability of molecular systems. Phys. Rev. A 51(2), 960–966 (1995)
[28] Revuz D.: Markov Chains. North–Holland, Amsterdam (1984) · Zbl 0539.60073
[29] Turinici G., Rabitz H.: Quantum wavefunction controllability. Chem. Phys. 267, 1–9 (2001) · Zbl 1064.81558
[30] Wang W.-M.: Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations. J. Func. Anal. 254(11), 2926–2946 (2008) · Zbl 1171.35029
[31] Wang W.-M.: Logarithmic bounds on Sobolev norms for time dependent linear Schrödinger equations. Commun. PDE 33(12), 2164–2179 (2008) · Zbl 1154.35450
[32] Zuazua E.: Remarks on the controllability of the Schrödinger equation. CRM Proc. Lecture Notes 33, 193–211 (2003)
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