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Some examples of generalized reflectionless Schrödinger potentials. (English) Zbl 1418.37115

Summary: The class of generalized reflectionless Schrödinger operators was introduced by D. Sh. Lundina [Teor. Funkts. Funkts. Anal. Prilozh. 44, 57–66 (1985; Zbl 0583.47046)] in 1985. V. A. Marchenko [Nonlinear equations and operator algebras. Transl. from the Russian by V. I. Rublinetsky. Dordrecht (Netherlands) etc.: D. Reidel Publishing Company (1988; Zbl 0644.47053)] worked out a useful parametrization of these potentials, and S. Kotani [Zh. Mat. Fiz. Anal. Geom. 4, No. 4, 490–528 (2008; Zbl 1183.35244)] showed that each such potential is of Sato-Segal-Wilson type. Nevertheless the dynamics under translation of a generic generalized reflectionless potential is still not well understood. We give examples which show that certain dynamical anomalies can occur.

MSC:

37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
35Q55 NLS equations (nonlinear Schrödinger equations)
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[1] J. Avron, Almost periodic Schrödinger operators II. The integrated density of states,, Duke Math. Jour., 50, 369 (1983) · Zbl 0544.35030 · doi:10.1215/S0012-7094-83-05016-0
[2] M. Bebutov, <em>On Dynamical Systems in the Space of Continuous Functions</em>,, Bull. Inst. Mat. Moskov. Gos. Univ. 2 (1940)., 2 (1940)
[3] E. Coddington, <em>Theory of Ordinary Differential Equations</em>,, Mc Graw-Hill (1955) · Zbl 0064.33002
[4] W. Coppel, <em>Dichotomies in Stability Theory</em>,, Lecture Notes in Mathematics (1978) · Zbl 0376.34001
[5] W. Craig, The trace formula for Schrödinger operators on the line,, Comm. Math. Phys., 126, 379 (1989) · Zbl 0681.34026 · doi:10.1007/BF02125131
[6] W. Craig, Subharmonicity of the Lyapunov index,, Duke Math. Jour., 50, 551 (1983) · Zbl 0518.35027 · doi:10.1215/S0012-7094-83-05025-1
[7] D. Damanik, Counterexamples to the Kotani-Last conjecture for continuum Schrödinger operators via character-automorphic Hardy spaces,, Adv. Math., 293, 738 (2016) · Zbl 1337.37037 · doi:10.1016/j.aim.2016.02.023
[8] C. De Concini, The algebraic-geometric AKNS potentials,, Ergod. Th. & Dynam. Sys., 7, 1 (1987) · Zbl 0636.35077 · doi:10.1017/S0143385700003783
[9] B. Dubrovin, Nonlinear equations of Korteweg-de Vries type, finite zone linear operators and Abelian varieties,, Russ. Math. Surveys, 31, 55 (1976) · Zbl 0326.35011
[10] P. Duren, <em>Theory of \(H^p\) Spaces</em>,, Academic Press (1970) · Zbl 0215.20203
[11] R. Ellis, <em>Lectures on Topological Dynamics</em>,, Benjamin (1969) · Zbl 0193.51502
[12] A. Eremenko, Comb functions,, Contemp. Math., 578, 99 (2012) · Zbl 1318.30011 · doi:10.1090/conm/578/11472
[13] F. Gesztesy, The xi function,, Acta Matematica, 176, 49 (1996) · Zbl 0885.34070 · doi:10.1007/BF02547335
[14] F. Gesztesy, Spectral properties of a class of reflectionless Schrödinger operators,, Jour. Func. Anal., 241, 486 (2006) · Zbl 1387.34120 · doi:10.1016/j.jfa.2006.08.006
[15] I. Goldsheid, A random homogeneous Schrödinger operator has pure point spectrum,, Funk. Anal. i Prilozh., 11, 1 (1977) · Zbl 0368.34015 · doi:10.1007/BF01135526
[16] M. Hasumi, <em>Hardy Classes on Infinitely Connected Riemann Surfaces</em>,, Lecture Notes in Math. 1027, 1027 (1983) · Zbl 0523.30028
[17] L. Helms, <em>Introduction to Potential Theory</em>,, Robert E. Krieger Publ. Co. (1975) · Zbl 0188.17203
[18] R. Johnson, The recurrent Hill’s equation,, Jour. Diff. Eqns, 46, 165 (1982) · Zbl 0535.34021 · doi:10.1016/0022-0396(82)90114-0
[19] R. Johnson, A review of recent work on almost periodic differential and difference operators,, Acta Applicandae Mathematicae, 1, 241 (1983) · Zbl 0542.34007 · doi:10.1007/BF00046601
[20] R. Johnson, Exponential dichotomy, rotation number and linear differential equations with bounded coefficients,, Jour. Diff. Eqns., 61, 54 (1986) · Zbl 0608.34056 · doi:10.1016/0022-0396(86)90125-7
[21] R. Johnson, Lyapunov numbers for the almost-periodic Schroedinger equation,, Illinois Jour. Math., 28, 397 (1984)
[22] R. Johnson, The rotation number for almost periodic potentials,, Comm. Math. Phys., 84, 403 (1982) · Zbl 0497.35026 · doi:10.1007/BF01208484
[23] R. Johnson, Some remarks concerning reflectionless Sturm-Liouville potentials,, Stoch. and Dynamics, 8, 413 (2008) · Zbl 1162.34021 · doi:10.1142/S0219493708002391
[24] R. Johnson, Remarks on a paper of Kotani concerning generalized reflectionless Schrödinger potentials,, Discr. Cont. Dynam. Sys. B, 14, 559 (2010) · Zbl 1204.37016 · doi:10.3934/dcdsb.2010.14.559
[25] R. Johnson, Remarks on the generalized reflectionless Schrödinger potentials,, Jour. Dynam. Diff. Eqns., 1 (2015) · Zbl 1367.37018 · doi:10.1007/s10884-014-9424-8
[26] S. Kotani, Lyapunov indices determine absolutely continuous spectrum of stationary random Schrödinger operators,, Proc. Taniguchi Symp. SA, 219 (1985)
[27] S. Kotani, Generalized Floquet theory for stationary Schrödinger operators in one dimension,, Chaos Solitons and Fractals, 8, 1817 (1997) · Zbl 0936.34074 · doi:10.1016/S0960-0779(97)00042-8
[28] S. Kotani, KdV flow on generalized reflectionless Schrödinger potentials,, Jour. Math. Phys., 4, 490 (2008) · Zbl 1183.35244
[29] D. Lundina, Compactness of the set of reflectionless potentials,, Funk. Anal. i Prilozh., 44, 55 (1985)
[30] V. Marchenko, The Cauchy problem for the KdV equation with non-decreasing initial data,, in What is Integrability?, 273 (1991) · Zbl 0810.34090
[31] H. McKean, The spectrum of Hill’s equation,, Invent. Math., 30, 217 (1975) · Zbl 0319.34024 · doi:10.1007/BF01425567
[32] J. Moser, An example of a Schrödinger operator with almost periodic potential and nowhere dense spectrum,, Helv. Math. Acta, 56, 198 (1981) · Zbl 0477.34018 · doi:10.1007/BF02566210
[33] V. Nemytskii, <em>Qualitative Theory of Differential Equations</em>,, Princeton Univ. Press (1960) · Zbl 0089.29502
[34] V. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19, 197 (1968) · Zbl 0236.93034
[35] L. Pastur, Spectral properties of disordered systems in the one-body approximation,, Comm. Math. Phys., 75, 179 (1980) · Zbl 0429.60099 · doi:10.1007/BF01222516
[36] C. Remling, Topological properties of reflectionelss Jacobi matrices,, J. Approx. Theory, 168, 1 (2013) · Zbl 1272.47042 · doi:10.1016/j.jat.2012.12.009
[37] R. Sacker, Existence of dichotomies and invariant splittings for linear differential systems II,, Jour. Diff. Eqns, 22, 478 (1976) · Zbl 0339.58013 · doi:10.1016/0022-0396(76)90042-5
[38] M. Sato, Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold,, North-Holland Mathematics Studies, 81, 259 (1983) · Zbl 0528.58020 · doi:10.1016/S0304-0208(08)72096-6
[39] G. Segal, {Loop groups and equations of K-dV type,, Publ. IHES, 61, 5 (1985)}
[40] B. Simon, Almost periodic Schrödinger operators: A review,, Adv. Appl. Math., 3, 463 (1982) · Zbl 0545.34023 · doi:10.1016/S0196-8858(82)80018-3
[41] B. Simon, A new approach to inverse spectral theory I. Fundamental formalism,, Annals of Math., 150, 1029 (1999) · Zbl 0945.34013 · doi:10.2307/121061
[42] M. Sodin, Almost periodic Jacobi matrices with homogeneous spectrum, infinite dimensional Jacobi inversion, and Hardy spaces of character-automorphic functions,, Jour. Geom. Anal., 7, 387 (1997) · Zbl 1041.47502 · doi:10.1007/BF02921627
[43] M. Sodin, Almost periodic Schrödinger operators with Cantor homogeneous spectrum,, Comment. Math. Helv., 70, 639 (1995) · Zbl 0846.34024 · doi:10.1007/BF02566026
[44] H. Weyl, Über gewöhnliche lineare Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen,, Math. Annalen, 68, 220 (1910) · JFM 41.0343.01 · doi:10.1007/BF01474161
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