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

Some examples of generalized reflectionless Schrödinger potentials

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

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.
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.

[2]

M. Bebutov, On Dynamical Systems in the Space of Continuous Functions,, Bull. Inst. Mat. Moskov. Gos. Univ. 2 (1940)., 2 (1940).

[3]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations,, Mc Graw-Hill, (1955).

[4]

W. Coppel, Dichotomies in Stability Theory,, Lecture Notes in Mathematics, (1978).

[5]

W. Craig, The trace formula for Schrödinger operators on the line,, Comm. Math. Phys., 126 (1989), 379. doi: 10.1007/BF02125131.

[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.

[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.

[8]

C. De Concini and R. Johnson, The algebraic-geometric AKNS potentials,, Ergod. Th. & Dynam. Sys., 7 (1987), 1. 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.

[10]

P. Duren, Theory of $H^p$ Spaces,, Academic Press, (1970).

[11]

R. Ellis, Lectures on Topological Dynamics,, Benjamin, (1969).

[12]

A. Eremenko and P. Yuditskii, Comb functions,, Contemp. Math., 578 (2012), 99. doi: 10.1090/conm/578/11472.

[13]

F. Gesztesy and B. Simon, The xi function,, Acta Matematica, 176 (1996), 49. 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. 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. doi: 10.1007/BF01135526.

[16]

M. Hasumi, Hardy Classes on Infinitely Connected Riemann Surfaces,, Lecture Notes in Math. 1027, 1027 (1983).

[17]

L. Helms, Introduction to Potential Theory,, Robert E. Krieger Publ. Co., (1975).

[18]

R. Johnson, The recurrent Hill's equation,, Jour. Diff. Eqns, 46 (1982), 165. 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. doi: 10.1007/BF00046601.

[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.

[21]

R. Johnson, Lyapunov numbers for the almost-periodic Schroedinger equation,, Illinois Jour. Math., 28 (1984), 397.

[22]

R. Johnson and J. Moser, The rotation number for almost periodic potentials,, Comm. Math. Phys., 84 (1982), 403. doi: 10.1007/BF01208484.

[23]

R. Johnson and L. Zampogni, Some remarks concerning reflectionless Sturm-Liouville potentials,, Stoch. and Dynamics, 8 (2008), 413. 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. 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. 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, (1985), 219.

[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.

[28]

S. Kotani, KdV flow on generalized reflectionless Schrödinger potentials,, Jour. Math. Phys., 4 (2008), 490.

[29]

D. Lundina, Compactness of the set of reflectionless potentials,, Funk. Anal. i Prilozh., 44 (1985), 55.

[30]

V. Marchenko, The Cauchy problem for the KdV equation with non-decreasing initial data,, in What is Integrability?, (1991), 273.

[31]

H. McKean and P. van Moerbeke, The spectrum of Hill's equation,, Invent. Math., 30 (1975), 217. 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. doi: 10.1007/BF02566210.

[33]

V. Nemytskii and V. Stepanov, Qualitative Theory of Differential Equations,, Princeton Univ. Press, (1960).

[34]

V. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197.

[35]

L. Pastur, Spectral properties of disordered systems in the one-body approximation,, Comm. Math. Phys., 75 (1980), 179. doi: 10.1007/BF01222516.

[36]

C. Remling, Topological properties of reflectionelss Jacobi matrices,, J. Approx. Theory, 168 (2013), 1. 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. 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. 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.

[40]

B. Simon, Almost periodic Schrödinger operators: A review,, Adv. Appl. Math., 3 (1982), 463. 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. 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. doi: 10.1007/BF02921627.

[43]

M. Sodin and P. Yuditskii, Almost periodic Schrödinger operators with Cantor homogeneous spectrum,, Comment. Math. Helv., 70 (1995), 639. doi: 10.1007/BF02566026.

[44]

H. Weyl, Über gewöhnliche lineare Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen,, Math. Annalen, 68 (1910), 220. 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. 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)., 2 (1940).

[3]

E. Coddington and N. Levinson, Theory of Ordinary Differential Equations,, Mc Graw-Hill, (1955).

[4]

W. Coppel, Dichotomies in Stability Theory,, Lecture Notes in Mathematics, (1978).

[5]

W. Craig, The trace formula for Schrödinger operators on the line,, Comm. Math. Phys., 126 (1989), 379. doi: 10.1007/BF02125131.

[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.

[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.

[8]

C. De Concini and R. Johnson, The algebraic-geometric AKNS potentials,, Ergod. Th. & Dynam. Sys., 7 (1987), 1. 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.

[10]

P. Duren, Theory of $H^p$ Spaces,, Academic Press, (1970).

[11]

R. Ellis, Lectures on Topological Dynamics,, Benjamin, (1969).

[12]

A. Eremenko and P. Yuditskii, Comb functions,, Contemp. Math., 578 (2012), 99. doi: 10.1090/conm/578/11472.

[13]

F. Gesztesy and B. Simon, The xi function,, Acta Matematica, 176 (1996), 49. 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. 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. doi: 10.1007/BF01135526.

[16]

M. Hasumi, Hardy Classes on Infinitely Connected Riemann Surfaces,, Lecture Notes in Math. 1027, 1027 (1983).

[17]

L. Helms, Introduction to Potential Theory,, Robert E. Krieger Publ. Co., (1975).

[18]

R. Johnson, The recurrent Hill's equation,, Jour. Diff. Eqns, 46 (1982), 165. 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. doi: 10.1007/BF00046601.

[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.

[21]

R. Johnson, Lyapunov numbers for the almost-periodic Schroedinger equation,, Illinois Jour. Math., 28 (1984), 397.

[22]

R. Johnson and J. Moser, The rotation number for almost periodic potentials,, Comm. Math. Phys., 84 (1982), 403. doi: 10.1007/BF01208484.

[23]

R. Johnson and L. Zampogni, Some remarks concerning reflectionless Sturm-Liouville potentials,, Stoch. and Dynamics, 8 (2008), 413. 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. 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. 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, (1985), 219.

[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.

[28]

S. Kotani, KdV flow on generalized reflectionless Schrödinger potentials,, Jour. Math. Phys., 4 (2008), 490.

[29]

D. Lundina, Compactness of the set of reflectionless potentials,, Funk. Anal. i Prilozh., 44 (1985), 55.

[30]

V. Marchenko, The Cauchy problem for the KdV equation with non-decreasing initial data,, in What is Integrability?, (1991), 273.

[31]

H. McKean and P. van Moerbeke, The spectrum of Hill's equation,, Invent. Math., 30 (1975), 217. 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. doi: 10.1007/BF02566210.

[33]

V. Nemytskii and V. Stepanov, Qualitative Theory of Differential Equations,, Princeton Univ. Press, (1960).

[34]

V. Oseledets, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197.

[35]

L. Pastur, Spectral properties of disordered systems in the one-body approximation,, Comm. Math. Phys., 75 (1980), 179. doi: 10.1007/BF01222516.

[36]

C. Remling, Topological properties of reflectionelss Jacobi matrices,, J. Approx. Theory, 168 (2013), 1. 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. 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. 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.

[40]

B. Simon, Almost periodic Schrödinger operators: A review,, Adv. Appl. Math., 3 (1982), 463. 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. 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. doi: 10.1007/BF02921627.

[43]

M. Sodin and P. Yuditskii, Almost periodic Schrödinger operators with Cantor homogeneous spectrum,, Comment. Math. Helv., 70 (1995), 639. doi: 10.1007/BF02566026.

[44]

H. Weyl, Über gewöhnliche lineare Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen,, Math. Annalen, 68 (1910), 220. doi: 10.1007/BF01474161.

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