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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

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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