Reducibility for Schrödinger Operator with Finite Smooth and Time-Quasi-periodic Potential

Abstract

In this paper, the author establishes a reduction theorem for linear Schrödinger equation with finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition by means of KAM (Kolmogorov-Arnold-Moser) technique. Moreover, it is proved that the corresponding Schrödinger operator possesses the property of pure point spectra and zero Lyapunov exponent.

Keywords

Reducibility / Quasi-periodic Schrödinger operator / KAM theory / Finite smooth potential / Lyapunov exponent / Pure-Point spectrum

References

[1] Bambusi D. Reducibility of 1-d Schrödinger equation with time quasi-periodic unbounded perturbations, II. Commun. Math. Phys., 2017, 353: 353-378

[2] Bambusi D. Reducibility of 1-d Schrödinger equation with time quasi-periodic unbounded perturbations, I. Trans. Amer. Math. Soc, 2018, 370: 1823-1865

[3] Bambusi D, Graffi S. Time quasi-periodic unbounded perturbations of schrödinger operators and KAM methods. Commun. Math. Phys, 2001, 219: 465-480

[4] Bambusi D, Grébert B, Maspero A, Robert D. Reducibility of the quantum harmonic oscillator in d-dimensions with polynomial time dependent perturbation. Analysis and PDE, 2017, 11(3): 775-799

[5] Bogoljubov N N, Mitropoliskii Ju A, Samoïlenko A M. Methods of Accelerated Convergence in Nonlinear Mechanics, 1976, New York: Springer-Verlag

[6] Bourgain J, Goldstein M. On nonperturbative localization with quasi-periodic potential. Ann. of Math., 2000, 152: 835-879

[7] Chierchia L, Qian D. Mosers theorem for lower dimensional tori. J. Diff. Eqs., 2004, 206: 55-93

[8] Combescure M. The quantum stability problem for time-periodic perturbations of the harmonic oscillator. Ann. Inst. Henri Poincaré, 1987, 47(1): 63-83

[9] Duclos P, Lev O, Stovicek P, Vittot M. Perturbation of an eigen-value from a dense point spectrum: A general Floquet Hamiltonian. Ann. Inst. Henri Poincaré, 1999, 71(3): 241-301

[10] Duclos P, Stovicek P. Floquet Hamiltonians with pure point spectrum. Coram. Math. Phys., 1996, 177(2): 327-347

[11] Eliasson H L, Kuksin S B. On reducibility of Schrödinger equations with quasi-periodic in time potentials. Gomm. Math. Phys., 2009, 286(1): 125-135

[12] Klein S. Anderson localization for the discrete one-dimensional quasi-periodic Schrödinger operator with potential defined by a Gevrey-class function. J. Fund. Anal, 2005, 218(2): 255-292

[13] Kuksin S B. Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Math, 1993, New York: Springer-Verlag 1556

[14] Liang, Z. and Wang, X., On reducibility of Id wave equation with quasi-periodic in time potentials, J. Dyn. Diff. Equat., DOI: 10.1007/s10884-017-9576-4.

[15] Liang Z, Wang Z. Reducibility of quantum harmonic oscillator on R^d with differential and quasi-periodic in time potential, 2017

[16] Pöschel J. A KAM-theorem for some nonlinear partial differential equations. Ann. Scuola Norm,. Sup. Pisa, CI. Sci., IV Ser., 1996, 23: 119-148

[17] Salamon D. The Kolmogorov-Arnold-Moser theorem. Mathematical Physics Electronic Journal, 2004, 10(3): 1-37

[18] Salamon D, Zehnder E. KAM theory in configuration space. Commun. Math. Helv., 1989, 64: 84-132

[19] Wang Z, Liang Z. Reducibility of ID quantum harmonic oscillator perturbed by a quasi-periodic potential with logarithmic decay. Nonlinearity, 2017, 30: 1405-1448

[20] Yuan X, Zhang K. A reduction theorem for time dependent Schrödinger operator with finite differ-entiable unbounded perturbation. J. Math. Phys., 2013, 54(5): 465-480