不连续斜积流的唯一遍历定理和线性薛定谔方程的旋转数
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摘要
传统的动力系统理论源自于经典力学模型,其主要的特征是这些动力系统具有时间和空间上的连续性乃至可微性。本文以具有量子效应的线性薛定谔方程为原始模型,提出了具有时空不连续性的斜积拟流的动力系统概念。为了研究方程的动力学行为,我们建立了相应的遍历性结论。全文分成理论和应用两部分。主要结论总结在第一章。
与传统斜积流比较,对于斜积拟流,不连续性是一个本质性的不同。为克服不连续性带来的困难,在第二章中,我们首先利用不动点定理和测度的收敛性质,建立了相应的不变测度存在性的定理。粗略地讲,只要不连续点不足够多,则斜积拟流仍然存在不变Borel概率测度。此结果是经典的Bogoliubov-Krylov定理的一个推广。再利用测度论和拓扑学的相关知识,我们建立了相应的唯一遍历定理。于是,对于斜积拟流的Birkho?时间平均,我们也可以得到一致收敛性结果。此结果是Johnson和Moser的唯一遍历定理的一个推广。
在第三章中,我们从紧致性的角度,对一维几乎周期格点给出了几个等价的刻画,并分析了一些动力学性质。为接下来的应用部分做好了铺垫工作。
在第四章中,作为斜积拟流的理论成果的一个重要应用,我们对具有几乎周期位势并且在几乎周期格点上具有相变的线性薛定谔方程建立了旋转数的概念。其中一个关键步骤就是,为了推导出相应的斜积拟流,我们不仅考虑了势能函数和相变矩阵,而且把格点作为斜积拟流的相空间元素也考虑了进去。相空间的扩充是建立旋转数概念过程中的一个关键突破口。同时,我们还部分地讨论了旋转数对于量子效应的强度参数的依赖关系。由此看出,目前这些工作已为更深入地研究具有量子效应的系统的性质奠定了重要的基础。
Models from classical mechanics are basic examples of dynamical systems. Oneof the main features is that these dynamical systems are continuous in temporal and spa-tial variables and are even di?erentiable. In this paper, motivated by linear Schro¨dingerequations with quantum e?ects, we will introduce the concept of skew-product quasi-?ows (SPQFs) which have both temporal and spatial discontinuity. To study the dy-namics of these equations, we will establish some ergodic results for SPQFs. Thispaper is divided into two parts: theory and application. Our main results are summa-rized in Chapter 1.
Compared with classical skew-product ?ows, the discontinuity of SPQFs is anessential di?erence. In Chapter 2, in order to overcome the di?culties coming fromthe discontinuity, we firstly apply the fixed point theorem and the weak convergence ofmeasures to establish the existence theorem of invariant measures of SPQFs. Roughlyspeaking, when the set of discontinuous points is small, SPQFs still admit some in-variant Borel probability measures. Such a result is an extension of the celebratedBogoliubov-Krylov theorem. Applying further results from measure theory and topol-ogy, we will establish the unique ergodic theorem for SPQFs, which yields a uniformconvergence for the Birkho? temporal average. This is a complete extension of theunique ergodic theorem of Johnson and Moser.
In Chapter 3, from point of view of compactness, we will give several equivalentdefinitions of almost periodic lattices, and analyze some dynamical properties corre-spondingly. This work plays a fundamental role in the following application.
In Chapter 4, as an application of theory of SPQFs, we will establish the concept ofrotation numbers for linear Schro¨dinger equations with almost periodic potentials andphase transmissions on almost periodic lattices. One of the crucial steps is that, in orderto yield the corresponding SPQFs, besides the potential and the phase transmission, weadd the lattice into the phase space of SPQFs. Such an extension of phase spaces is a crucial breakthrough to establish rotation numbers. Meanwhile, we partially discussthe dependence of rotation numbers on the intensive parameter of quantum e?ects. Inour opinion, the present work has laid an important foundation for further study ofthese systems with quantum e?ects.
与传统斜积流比较,对于斜积拟流,不连续性是一个本质性的不同。为克服不连续性带来的困难,在第二章中,我们首先利用不动点定理和测度的收敛性质,建立了相应的不变测度存在性的定理。粗略地讲,只要不连续点不足够多,则斜积拟流仍然存在不变Borel概率测度。此结果是经典的Bogoliubov-Krylov定理的一个推广。再利用测度论和拓扑学的相关知识,我们建立了相应的唯一遍历定理。于是,对于斜积拟流的Birkho?时间平均,我们也可以得到一致收敛性结果。此结果是Johnson和Moser的唯一遍历定理的一个推广。
在第三章中,我们从紧致性的角度,对一维几乎周期格点给出了几个等价的刻画,并分析了一些动力学性质。为接下来的应用部分做好了铺垫工作。
在第四章中,作为斜积拟流的理论成果的一个重要应用,我们对具有几乎周期位势并且在几乎周期格点上具有相变的线性薛定谔方程建立了旋转数的概念。其中一个关键步骤就是,为了推导出相应的斜积拟流,我们不仅考虑了势能函数和相变矩阵,而且把格点作为斜积拟流的相空间元素也考虑了进去。相空间的扩充是建立旋转数概念过程中的一个关键突破口。同时,我们还部分地讨论了旋转数对于量子效应的强度参数的依赖关系。由此看出,目前这些工作已为更深入地研究具有量子效应的系统的性质奠定了重要的基础。
Models from classical mechanics are basic examples of dynamical systems. Oneof the main features is that these dynamical systems are continuous in temporal and spa-tial variables and are even di?erentiable. In this paper, motivated by linear Schro¨dingerequations with quantum e?ects, we will introduce the concept of skew-product quasi-?ows (SPQFs) which have both temporal and spatial discontinuity. To study the dy-namics of these equations, we will establish some ergodic results for SPQFs. Thispaper is divided into two parts: theory and application. Our main results are summa-rized in Chapter 1.
Compared with classical skew-product ?ows, the discontinuity of SPQFs is anessential di?erence. In Chapter 2, in order to overcome the di?culties coming fromthe discontinuity, we firstly apply the fixed point theorem and the weak convergence ofmeasures to establish the existence theorem of invariant measures of SPQFs. Roughlyspeaking, when the set of discontinuous points is small, SPQFs still admit some in-variant Borel probability measures. Such a result is an extension of the celebratedBogoliubov-Krylov theorem. Applying further results from measure theory and topol-ogy, we will establish the unique ergodic theorem for SPQFs, which yields a uniformconvergence for the Birkho? temporal average. This is a complete extension of theunique ergodic theorem of Johnson and Moser.
In Chapter 3, from point of view of compactness, we will give several equivalentdefinitions of almost periodic lattices, and analyze some dynamical properties corre-spondingly. This work plays a fundamental role in the following application.
In Chapter 4, as an application of theory of SPQFs, we will establish the concept ofrotation numbers for linear Schro¨dinger equations with almost periodic potentials andphase transmissions on almost periodic lattices. One of the crucial steps is that, in orderto yield the corresponding SPQFs, besides the potential and the phase transmission, weadd the lattice into the phase space of SPQFs. Such an extension of phase spaces is a crucial breakthrough to establish rotation numbers. Meanwhile, we partially discussthe dependence of rotation numbers on the intensive parameter of quantum e?ects. Inour opinion, the present work has laid an important foundation for further study ofthese systems with quantum e?ects.
引文
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[3] Arnold L. Random Dynamical Systems. Springer-Verlag, Berlin, 1998.
[4] Avila A, Krikorian R. Reducibility or nonuniform hyperbolicity for quasiperiodicSchro¨dinger cocycles. Ann. of Math. (2), 2006, 164: 911–940.
[5] Avron J, Simon B. Almost periodic Schro¨dinger operators, I. Limit periodic potentials.Comm. Math. Phys., 1981, 82: 101–120.
[6] Avron J, Simon B. Almost periodic Schro¨dinger operators, II. The integrated density ofstates. Duke Math. J., 1983, 50: 369–391.
[7] Barreira L, Pesin Y. Lectures on Lyapunov Exponents and Smooth Ergodic Theory.American Mathematical Society, Providence RI, 2002.
[8] Binding P L, Rynne B P. Half-eigenvalues of periodic Sturm-Liouville problems. J. Dif-ferential Equations, 2004, 206: 280–305.
[9] Binding P L, Rynne B P. The spectrum of the periodic p-Laplacian. J. Di?erential Equa-tions, 2007, 235: 199–218.
[10] Binding P L, Rynne B P. Variational and non-variational eigenvalues of the p-Laplacian.J. Di?erential Equations, 2008, 244: 24–39.
[11] Birkho? G D. Dynamical Systems. American Mathematical Society, Providence RI,1927.
[12] Birkho? G D. Proof of the ergodic theorem. Proc. Nat. Acad. Sci. USA, 1931, 17: 656–660.
[13] Brin M, Stuck G. Introduction to Dynamical Systems. Cambridge Univ. Press, Cam-bridge, 2002.
[14] Boshernitzan M, Damanik D. Generic continuous spectrum for ergodic Schro¨dinger op-erators. Comm. Math. Phys., 2008, 283: 647–662.
[15] Cao Y. On growth rates of sub-additive functions for semi-?ow: Determined and randomcases. J. Di?erential Equations, 2006, 231: 1–17.
[16] Crauel H. Random Probability Measures on Polish Spaces. Taylor-Francis, London/NewYork, 2002.
[17] Colonius F, Fabbri R, Johnson R. Chain recurrence, growth rates and ergodic limits.Ergod. Th. Dynam. Syst., 2007, 27: 1509–1524.
[18] Dai X. Liao style numbers of di?erential systems. Commun. Contemp. Math., 2004, 6:279-299.
[19] Dai X. Integral expressions of Lyapunov exponents for autonomous ordinary di?erentialsystems. Sci. China Ser. A, 2009, 52: 195–216.
[20] Damanik D. Lyapunov exponents and spectral analysis of ergodic Schro¨dinger opera-tors: a survey of Kotani theory and its applications, in“Spectral theory and mathemati-cal physics: a Festschrift in honor of Barry Simon’s 60th birthday”. pp. 539–563, Proc.Sympos. Pure Math., 76, Part 2, Amer. Math. Soc., Providence, RI, 2007.
[21] Deift P, Simon B. Almost periodic Schro¨dinger operators, III. The absolutely continuousspectrum in one dimension. Comm. Math. Phys., 1983, 90: 389–411.
[22] Dieudonne′J. Foundations of Modern Analysis. Academic Press, New York/London,1960.
[23] Dunford N, Schwartz J T. Linear Operators, Part I. Interscience Publishers, New York,1958.
[24] Eliasson L H. Floquet solutions for the one-dimensional quasi-periodic Schro¨dingerequation. Comm. Math. Phys., 1992, 146: 447–482.
[25] Eliasson L H. Discrete one-dimensional quasi-periodic Schro¨dinger operators with purepoint spectrum. Acta Math., 1997, 179: 153–196.
[26] Fabbri R, Johnson R, Nu′n?ez C. Rotation number for non-autonomous linear Hamilto-nian systems I: Basic properties. Z. Angew. Math. Phys., 2003, 54: 484–502.
[27] Fayad B, Krikorian R. Exponential growth of product of matrices in SL(2, R). Nonlin-earity, 2008, 21:319–323.
[28] Feng D. Lyapunov exponents for products of matrices and multifractal analysis, I. Posi-tive matrices. Israel J. Math., 2003, 138: 353–376.
[29] Feng H, Zhang M. Optimal estimates on rotation number of almost periodic systems. Z.Angew. Math. Phys., 2006, 57: 183–204.
[30] Fink A. Almost Periodic Di?erential Equations. Springer, New York/Berlin, 1974.
[31] Franks J. Rotation numbers and instability sets. Bull. Amer. Math. Soc., 2003, 40: 263–279.
[32] Hale J K. Ordinary Di?erential Equations. second ed., Krieger, New York, 1980.
[33] Halmos P R. Measure Theory. Springer-Verlag, New York/Berlin, 1974.
[34] Her H, You J. Full measure reducibility for generic one-parameter family of quasi-periodic linear systems. J. Dynam. Di?erential Equations, 2008, 20: 831–866.
[35] Herman M. A method for determining the lower bounds of Lyapunov exponents andsome examples showing the local character of an Arnold-Moser theorem on the two-dimensional torus. Comment. Math. Helv., 1983, 58: 453-502. (in French)
[36] Host B, Kra B. Non-conventional ergodic averages and nilmanifolds. Ann. Math. (2),2005, 161: 397–488.
[37] Hou X, You J. The rigidity of reducibility of cocycles on SO(N, R). Nonlinearity, 2008,21: 2317–2330.
[38] Huang W, Yi Y. Almost periodically forced circle ?ows. J. Funct. Anal., 2009, 257:832–902.
[39] Johnson R, Moser J. The rotation number for almost periodic potentials. Comm. Math.Phys., 1982, 84: 403–438; Erratum. Comm. Math. Phys., 1983, 90: 317–318.
[40] Johnson R A, Palmer K J, Sell G R. Ergodic properties of linear dynamical systems.SIAM J. Math. Anal., 1987, 18: 1–33.
[41] Kwapisz J. Poincare′rotation number for maps of the real line with almost periodicdisplacement. Nonlinearity, 2000, 13: 1841–1854.
[42] Katok A, Hasselblatt B. Introduction to the Modern Theory of Dynamical Systems.Cambridge Univ. Press, Cambridge, 1995.
[43] Kingman J F C. Subadditive ergodic theory. Ann. Probability, 1973, 1: 883–909.
[44] Kronig R, Penney W. Quantum mechanics in crystal lattices. Proc. Royal Soc. London,1931, 130: 499–513.
[45] Liang C, Liu G, Sun W. Approximation properties on invariant measure and oseledecsplitting in non-uniformly hyperbolic systems. Trans. Amer. Math. Soc., 2009, 361:1543–1579.
[46] Lenz D, Stollmann P. Quasicrystals, aperiodic order, and groupoid von Neumann alge-bras. C. R. Math. Acad. Sci. Paris, 2002, 334: 1131–1136.
[47] Lenz D, Stollmann P. An ergodic theorem for Delone dynamical systems and existenceof the integrated density of states. J. Anal. Math., 2005, 97: 1–24.
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