-
Previous Article
Local study of a renormalization operator for 1D maps under quasiperiodic forcing
- DCDS-S Home
- This Issue
-
Next Article
On the nonautonomous Hopf bifurcation problem
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 |
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. |
[1] |
Russell Johnson, Luca Zampogni. Remarks on a paper of Kotani concerning generalized reflectionless Schrödinger potentials. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 559-586. doi: 10.3934/dcdsb.2010.14.559 |
[2] |
Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940 |
[3] |
Victor Isakov. Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 631-640. doi: 10.3934/dcdss.2011.4.631 |
[4] |
Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial and Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259 |
[5] |
Victor Isakov, Jenn-Nan Wang. Increasing stability for determining the potential in the Schrödinger equation with attenuation from the Dirichlet-to-Neumann map. Inverse Problems and Imaging, 2014, 8 (4) : 1139-1150. doi: 10.3934/ipi.2014.8.1139 |
[6] |
Sebastián Ferrer, Francisco Crespo. Alternative angle-based approach to the $\mathcal{KS}$-Map. An interpretation through symmetry and reduction. Journal of Geometric Mechanics, 2018, 10 (3) : 359-372. doi: 10.3934/jgm.2018013 |
[7] |
Hyun-Jung Kim. Stochastic parabolic Anderson model with time-homogeneous generalized potential: Mild formulation of solution. Communications on Pure and Applied Analysis, 2019, 18 (2) : 795-807. doi: 10.3934/cpaa.2019038 |
[8] |
Hongxia Shi, Haibo Chen. Infinitely many solutions for generalized quasilinear Schrödinger equations with sign-changing potential. Communications on Pure and Applied Analysis, 2018, 17 (1) : 53-66. doi: 10.3934/cpaa.2018004 |
[9] |
Miaohua Jiang. Derivative formula of the potential function for generalized SRB measures of hyperbolic systems of codimension one. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 967-983. doi: 10.3934/dcds.2015.35.967 |
[10] |
Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure and Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63 |
[11] |
Yanheng Ding, Fukun Zhao. On a diffusion system with bounded potential. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 1073-1086. doi: 10.3934/dcds.2009.23.1073 |
[12] |
Farman Mamedov, Sara Monsurrò, Maria Transirico. Potential estimates and applications to elliptic equations. Conference Publications, 2015, 2015 (special) : 793-800. doi: 10.3934/proc.2015.0793 |
[13] |
Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068 |
[14] |
Zifei Shen, Fashun Gao, Minbo Yang. On critical Choquard equation with potential well. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3567-3593. doi: 10.3934/dcds.2018151 |
[15] |
Yang Liu, Wenke Li. A family of potential wells for a wave equation. Electronic Research Archive, 2020, 28 (2) : 807-820. doi: 10.3934/era.2020041 |
[16] |
Manseob Lee, Jumi Oh, Xiao Wen. Diffeomorphisms with a generalized Lipschitz shadowing property. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1913-1927. doi: 10.3934/dcds.2020346 |
[17] |
Shair Ahmad, Alan C. Lazer. On a property of a generalized Kolmogorov population model. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 1-6. doi: 10.3934/dcds.2013.33.1 |
[18] |
Alberto Maspero, Beat Schaad. One smoothing property of the scattering map of the KdV on $\mathbb{R}$. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1493-1537. doi: 10.3934/dcds.2016.36.1493 |
[19] |
Yoshikazu Giga, Hirotoshi Kuroda. A counterexample to finite time stopping property for one-harmonic map flow. Communications on Pure and Applied Analysis, 2015, 14 (1) : 121-125. doi: 10.3934/cpaa.2015.14.121 |
[20] |
Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]