Some examples of generalized reflectionless Schrödinger potentials

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August 2016, 9(4): 1149-1170. doi: 10.3934/dcdss.2016046

Some examples of generalized reflectionless Schrödinger potentials

Russell Johnson ^1, and Luca Zampogni ^2,

1.	Dipartimento di Matematica e Informatica, Università di Firenze, Via di Santa Marta 3, 50139 Firenze
2.	Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, Italy

Received May 2015 Revised October 2015 Published August 2016

Abstract
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the class of generalized reflectionless Schrödinger operators was introduced by Lundina in 1985. Marchenko worked out a useful parametrization of these potentials, and Kotani 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.

Keywords: reflectionless potential in the sense of Craig, glueing property, divisor map., Generalized reflectionless potential.

Mathematics Subject Classification: 37B55, 34B20, 34L40, 31A3.

Citation: Russell Johnson, Luca Zampogni. Some examples of generalized reflectionless Schrödinger potentials. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1149-1170. doi: 10.3934/dcdss.2016046

References:

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

show all references

References:

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

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