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 and 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), 369-391. doi: 10.1215/S0012-7094-83-05016-0.
[2]	M. Bebutov, On Dynamical Systems in the Space of Continuous Functions, Bull. Inst. Mat. Moskov. Gos. Univ. 2 (1940).
[3]	E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Mc Graw-Hill, New York, 1955.
[4]	W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629. Springer-Verlag, Berlin-New York, 1978.
[5]	W. Craig, The trace formula for Schrödinger operators on the line, Comm. Math. Phys., 126 (1989), 379-407. doi: 10.1007/BF02125131.
[6]	W. Craig and B. Simon, Subharmonicity of the Lyapunov index, Duke Math. Jour., 50 (1983), 551-560. doi: 10.1215/S0012-7094-83-05025-1.
[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), 738-781, arXiv:1405.6342. doi: 10.1016/j.aim.2016.02.023.
[8]	C. De Concini and R. Johnson, The algebraic-geometric AKNS potentials, Ergod. Th. & Dynam. Sys., 7 (1987), 1-24. doi: 10.1017/S0143385700003783.
[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), 55-136.
[10]	P. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
[11]	R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.
[12]	A. Eremenko and P. Yuditskii, Comb functions, Contemp. Math., 578 (2012), 99-118. doi: 10.1090/conm/578/11472.
[13]	F. Gesztesy and B. Simon, The xi function, Acta Matematica, 176 (1996), 49-71. doi: 10.1007/BF02547335.
[14]	F. Gesztesy and P. Yuditskii, Spectral properties of a class of reflectionless Schrödinger operators, Jour. Func. Anal., 241 (2006), 486-527. doi: 10.1016/j.jfa.2006.08.006.
[15]	I. Goldsheid, S. Molchanov and L. Pastur, A random homogeneous Schrödinger operator has pure point spectrum, Funk. Anal. i Prilozh., 11 (1977), 1-10, 96. doi: 10.1007/BF01135526.
[16]	M. Hasumi, Hardy Classes on Infinitely Connected Riemann Surfaces, Lecture Notes in Math. 1027, Springer-Verlag, Berlin, 1983.
[17]	L. Helms, Introduction to Potential Theory, Robert E. Krieger Publ. Co., Huntington USA, 1975.
[18]	R. Johnson, The recurrent Hill's equation, Jour. Diff. Eqns, 46 (1982), 165-193. 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 (1983), 241-261. doi: 10.1007/BF00046601.
[20]	R. Johnson, Exponential dichotomy, rotation number and linear differential equations with bounded coefficients, Jour. Diff. Eqns., 61 (1986), 54-78. doi: 10.1016/0022-0396(86)90125-7.
[21]	R. Johnson, Lyapunov numbers for the almost-periodic Schroedinger equation, Illinois Jour. Math., 28 (1984), 397-419.
[22]	R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438. doi: 10.1007/BF01208484.
[23]	R. Johnson and L. Zampogni, Some remarks concerning reflectionless Sturm-Liouville potentials, Stoch. and Dynamics, 8 (2008), 413-449. doi: 10.1142/S0219493708002391.
[24]	R. Johnson and L. Zampogni, Remarks on a paper of Kotani concerning generalized reflectionless Schrödinger potentials, Discr. Cont. Dynam. Sys. B, 14 (2010), 559-586. doi: 10.3934/dcdsb.2010.14.559.
[25]	R. Johnson and L. Zampogni, Remarks on the generalized reflectionless Schrödinger potentials, Jour. Dynam. Diff. Eqns., (2015), 1-29. 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, Katata, (1985), 219-250.
[27]	S. Kotani, Generalized Floquet theory for stationary Schrödinger operators in one dimension, Chaos Solitons and Fractals, 8 (1997), 1817-1854. doi: 10.1016/S0960-0779(97)00042-8.
[28]	S. Kotani, KdV flow on generalized reflectionless Schrödinger potentials, Jour. Math. Phys., Anal., Geom., 4 (2008), 490-528, 574.
[29]	D. Lundina, Compactness of the set of reflectionless potentials, Funk. Anal. i Prilozh., 44 (1985), 55-66.
[30]	V. Marchenko, The Cauchy problem for the KdV equation with non-decreasing initial data, in What is Integrability?, Springer series in Nonlinear Dynamics, ed. V. Zakharov, Springer-Verlag, Berlin, 1991, 273-318.
[31]	H. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math., 30 (1975), 217-274. 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 (1981), 198-224. doi: 10.1007/BF02566210.
[33]	V. Nemytskii and V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press, Princeton, 1960.
[34]	V. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231.
[35]	L. Pastur, Spectral properties of disordered systems in the one-body approximation, Comm. Math. Phys., 75 (1980), 179-196. doi: 10.1007/BF01222516.
[36]	C. Remling, Topological properties of reflectionelss Jacobi matrices, J. Approx. Theory, 168 (2013), 1-17. doi: 10.1016/j.jat.2012.12.009.
[37]	R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems II, Jour. Diff. Eqns, 22 (1976), 478-496. 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 (1983), 259-271. doi: 10.1016/S0304-0208(08)72096-6.
[39]	G. Segal and G. Wilson, {Loop groups and equations of K-dV type, Publ. IHES, 61 (1985), 5-65.
[40]	B. Simon, Almost periodic Schrödinger operators: A review, Adv. Appl. Math., 3 (1982), 463-490. 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 (1999), 1029-1057. doi: 10.2307/121061.
[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), 387-435. doi: 10.1007/BF02921627.
[43]	M. Sodin and P. Yuditskii, Almost periodic Schrödinger operators with Cantor homogeneous spectrum, Comment. Math. Helv., 70 (1995), 639-658. doi: 10.1007/BF02566026.
[44]	H. Weyl, Über gewöhnliche lineare Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Annalen, 68 (1910), 220-269. doi: 10.1007/BF01474161.

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), 369-391. doi: 10.1215/S0012-7094-83-05016-0.
[2]	M. Bebutov, On Dynamical Systems in the Space of Continuous Functions, Bull. Inst. Mat. Moskov. Gos. Univ. 2 (1940).
[3]	E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, Mc Graw-Hill, New York, 1955.
[4]	W. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics, Vol. 629. Springer-Verlag, Berlin-New York, 1978.
[5]	W. Craig, The trace formula for Schrödinger operators on the line, Comm. Math. Phys., 126 (1989), 379-407. doi: 10.1007/BF02125131.
[6]	W. Craig and B. Simon, Subharmonicity of the Lyapunov index, Duke Math. Jour., 50 (1983), 551-560. doi: 10.1215/S0012-7094-83-05025-1.
[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), 738-781, arXiv:1405.6342. doi: 10.1016/j.aim.2016.02.023.
[8]	C. De Concini and R. Johnson, The algebraic-geometric AKNS potentials, Ergod. Th. & Dynam. Sys., 7 (1987), 1-24. doi: 10.1017/S0143385700003783.
[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), 55-136.
[10]	P. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
[11]	R. Ellis, Lectures on Topological Dynamics, Benjamin, New York, 1969.
[12]	A. Eremenko and P. Yuditskii, Comb functions, Contemp. Math., 578 (2012), 99-118. doi: 10.1090/conm/578/11472.
[13]	F. Gesztesy and B. Simon, The xi function, Acta Matematica, 176 (1996), 49-71. doi: 10.1007/BF02547335.
[14]	F. Gesztesy and P. Yuditskii, Spectral properties of a class of reflectionless Schrödinger operators, Jour. Func. Anal., 241 (2006), 486-527. doi: 10.1016/j.jfa.2006.08.006.
[15]	I. Goldsheid, S. Molchanov and L. Pastur, A random homogeneous Schrödinger operator has pure point spectrum, Funk. Anal. i Prilozh., 11 (1977), 1-10, 96. doi: 10.1007/BF01135526.
[16]	M. Hasumi, Hardy Classes on Infinitely Connected Riemann Surfaces, Lecture Notes in Math. 1027, Springer-Verlag, Berlin, 1983.
[17]	L. Helms, Introduction to Potential Theory, Robert E. Krieger Publ. Co., Huntington USA, 1975.
[18]	R. Johnson, The recurrent Hill's equation, Jour. Diff. Eqns, 46 (1982), 165-193. 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 (1983), 241-261. doi: 10.1007/BF00046601.
[20]	R. Johnson, Exponential dichotomy, rotation number and linear differential equations with bounded coefficients, Jour. Diff. Eqns., 61 (1986), 54-78. doi: 10.1016/0022-0396(86)90125-7.
[21]	R. Johnson, Lyapunov numbers for the almost-periodic Schroedinger equation, Illinois Jour. Math., 28 (1984), 397-419.
[22]	R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438. doi: 10.1007/BF01208484.
[23]	R. Johnson and L. Zampogni, Some remarks concerning reflectionless Sturm-Liouville potentials, Stoch. and Dynamics, 8 (2008), 413-449. doi: 10.1142/S0219493708002391.
[24]	R. Johnson and L. Zampogni, Remarks on a paper of Kotani concerning generalized reflectionless Schrödinger potentials, Discr. Cont. Dynam. Sys. B, 14 (2010), 559-586. doi: 10.3934/dcdsb.2010.14.559.
[25]	R. Johnson and L. Zampogni, Remarks on the generalized reflectionless Schrödinger potentials, Jour. Dynam. Diff. Eqns., (2015), 1-29. 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, Katata, (1985), 219-250.
[27]	S. Kotani, Generalized Floquet theory for stationary Schrödinger operators in one dimension, Chaos Solitons and Fractals, 8 (1997), 1817-1854. doi: 10.1016/S0960-0779(97)00042-8.
[28]	S. Kotani, KdV flow on generalized reflectionless Schrödinger potentials, Jour. Math. Phys., Anal., Geom., 4 (2008), 490-528, 574.
[29]	D. Lundina, Compactness of the set of reflectionless potentials, Funk. Anal. i Prilozh., 44 (1985), 55-66.
[30]	V. Marchenko, The Cauchy problem for the KdV equation with non-decreasing initial data, in What is Integrability?, Springer series in Nonlinear Dynamics, ed. V. Zakharov, Springer-Verlag, Berlin, 1991, 273-318.
[31]	H. McKean and P. van Moerbeke, The spectrum of Hill's equation, Invent. Math., 30 (1975), 217-274. 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 (1981), 198-224. doi: 10.1007/BF02566210.
[33]	V. Nemytskii and V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press, Princeton, 1960.
[34]	V. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc., 19 (1968), 197-231.
[35]	L. Pastur, Spectral properties of disordered systems in the one-body approximation, Comm. Math. Phys., 75 (1980), 179-196. doi: 10.1007/BF01222516.
[36]	C. Remling, Topological properties of reflectionelss Jacobi matrices, J. Approx. Theory, 168 (2013), 1-17. doi: 10.1016/j.jat.2012.12.009.
[37]	R. Sacker and G. Sell, Existence of dichotomies and invariant splittings for linear differential systems II, Jour. Diff. Eqns, 22 (1976), 478-496. 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 (1983), 259-271. doi: 10.1016/S0304-0208(08)72096-6.
[39]	G. Segal and G. Wilson, {Loop groups and equations of K-dV type, Publ. IHES, 61 (1985), 5-65.
[40]	B. Simon, Almost periodic Schrödinger operators: A review, Adv. Appl. Math., 3 (1982), 463-490. 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 (1999), 1029-1057. doi: 10.2307/121061.
[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), 387-435. doi: 10.1007/BF02921627.
[43]	M. Sodin and P. Yuditskii, Almost periodic Schrödinger operators with Cantor homogeneous spectrum, Comment. Math. Helv., 70 (1995), 639-658. doi: 10.1007/BF02566026.
[44]	H. Weyl, Über gewöhnliche lineare Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Annalen, 68 (1910), 220-269. doi: 10.1007/BF01474161.

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