## On the Kotani-Last and Schrödinger conjectures

HTML articles powered by AMS MathViewer

- by Artur Avila
- J. Amer. Math. Soc.
**28**(2015), 579-616 - DOI: https://doi.org/10.1090/S0894-0347-2014-00814-6
- Published electronically: June 11, 2014
- PDF | Request permission

## Abstract:

In the theory of ergodic one-dimensional Schrödinger operators, the ac spectrum has been traditionally expected to be very rigid. Two key conjectures in this direction state, on the one hand, that the ac spectrum demands almost periodicity of the potential, and, on the other hand, that the eigenfunctions are almost surely bounded in the essential support of the ac spectrum. We show how the repeated slow deformation of periodic potentials can be used to break rigidity, and disprove both conjectures.## References

- Artur Avila, Bassam Fayad, and Raphaël Krikorian,
*A KAM scheme for $\textrm {SL}(2,\Bbb R)$ cocycles with Liouvillean frequencies*, Geom. Funct. Anal.**21**(2011), no. 5, 1001–1019. MR**2846380**, DOI 10.1007/s00039-011-0135-6 - Joseph Avron and Barry Simon,
*Almost periodic Schrödinger operators. I. Limit periodic potentials*, Comm. Math. Phys.**82**(1981/82), no. 1, 101–120. MR**638515**, DOI 10.1007/BF01206947 - Michael Christ, Alexander Kiselev, and Christian Remling,
*The absolutely continuous spectrum of one-dimensional Schrödinger operators with decaying potentials*, Math. Res. Lett.**4**(1997), no. 5, 719–723. MR**1484702**, DOI 10.4310/MRL.1997.v4.n5.a9 - Corrado De Concini and Russell A. Johnson,
*The algebraic-geometric AKNS potentials*, Ergodic Theory Dynam. Systems**7**(1987), no. 1, 1–24. MR**886368**, DOI 10.1017/S0143385700003783 - David Damanik,
*Lyapunov exponents and spectral analysis of ergodic Schrödinger operators: a survey of Kotani theory and its applications*, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, Proc. Sympos. Pure Math., vol. 76, Amer. Math. Soc., Providence, RI, 2007, pp. 539–563. MR**2307747**, DOI 10.1090/pspum/076.2/2307747 - P. Deift and R. Killip,
*On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials*, Comm. Math. Phys.**203**(1999), no. 2, 341–347. MR**1697600**, DOI 10.1007/s002200050615 - P. Deift and B. Simon,
*Almost periodic Schrödinger operators. III. The absolutely continuous spectrum in one dimension*, Comm. Math. Phys.**90**(1983), no. 3, 389–411. MR**719297**, DOI 10.1007/BF01206889 - E. I. Dinaburg and Ja. G. Sinaĭ,
*The one-dimensional Schrödinger equation with quasiperiodic potential*, Funkcional. Anal. i Priložen.**9**(1975), no. 4, 8–21 (Russian). MR**0470318** - Bassam Fayad and Raphaël Krikorian,
*Rigidity results for quasiperiodic $\textrm {SL}(2,\Bbb R)$-cocycles*, J. Mod. Dyn.**3**(2009), no. 4, 497–510. MR**2587083**, DOI 10.3934/jmd.2009.3.479 - Svetlana Jitomirskaya,
*Ergodic Schrödinger operators (on one foot)*, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, Proc. Sympos. Pure Math., vol. 76, Amer. Math. Soc., Providence, RI, 2007, pp. 613–647. MR**2307750**, DOI 10.1090/pspum/076.2/2307750 - I. S. Kac,
*Spectral multiplicity of a second-order differential operator and expansion in eigenfunction*, Izv. Akad. Nauk SSSR Ser. Mat.**27**(1963), 1081–1112 (Russian). MR**0159982** - Shinichi Kotani,
*Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators*, Stochastic analysis (Katata/Kyoto, 1982) North-Holland Math. Library, vol. 32, North-Holland, Amsterdam, 1984, pp. 225–247. MR**780760**, DOI 10.1016/S0924-6509(08)70395-7 - S. Kotani,
*Generalized Floquet theory for stationary Schrödinger operators in one dimension*, Chaos Solitons Fractals**8**(1997), no. 11, 1817–1854. MR**1477262**, DOI 10.1016/S0960-0779(97)00042-8 - S. Kotani and M. Krishna,
*Almost periodicity of some random potentials*, J. Funct. Anal.**78**(1988), no. 2, 390–405. MR**943504**, DOI 10.1016/0022-1236(88)90125-5 - Yoram Last and Barry Simon,
*Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators*, Invent. Math.**135**(1999), no. 2, 329–367. MR**1666767**, DOI 10.1007/s002220050288 - Victor P. Maslov, Stanislav A. Molchanov, and Alexander Ya. Gordon,
*Behavior of generalized eigenfunctions at infinity and the Schrödinger conjecture*, Russian J. Math. Phys.**1**(1993), no. 1, 71–104. MR**1240494** - Camil Muscalu, Terence Tao, and Christoph Thiele,
*A Carleson theorem for a Cantor group model of the scattering transform*, Nonlinearity**16**(2003), no. 1, 219–246. MR**1950785**, DOI 10.1088/0951-7715/16/1/314 - Camil Muscalu, Terence Tao, and Christoph Thiele,
*A counterexample to a multilinear endpoint question of Christ and Kiselev*, Math. Res. Lett.**10**(2003), no. 2-3, 237–246. MR**1981900**, DOI 10.4310/MRL.2003.v10.n2.a10 - M. Reed,
*Functional Analysis*, Methods of Modern Mathematical Physics, Vol. 1, Academic Press, New York, NY, 1972. - Christian Remling,
*The absolutely continuous spectrum of Jacobi matrices*, Ann. of Math. (2)**174**(2011), no. 1, 125–171. MR**2811596**, DOI 10.4007/annals.2011.174.1.4 - Barry Simon,
*Equilibrium measures and capacities in spectral theory*, Inverse Probl. Imaging**1**(2007), no. 4, 713–772. MR**2350223**, DOI 10.3934/ipi.2007.1.713 - Walter Van Assche,
*Orthogonal polynomials and approximation theory: some open problems*, Recent trends in orthogonal polynomials and approximation theory, Contemp. Math., vol. 507, Amer. Math. Soc., Providence, RI, 2010, pp. 287–298. MR**2647574**, DOI 10.1090/conm/507/09965 - A. Volberg and P. Yuditskii,
*Kotani-Last problem and Hardy spaces on surfaces of Widom type*, J. Inventiones mathematicae (2013), 1–58. - Jean-Christophe Yoccoz,
*Some questions and remarks about $\textrm {SL}(2,\mathbf R)$ cocycles*, Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004, pp. 447–458. MR**2093316** - S. Ya. Zhitomirskaya,
*Singular spectral properties of a one-dimensional Schrödinger operator with almost periodic potential*, Dynamical systems and statistical mechanics (Moscow, 1991) Adv. Soviet Math., vol. 3, Amer. Math. Soc., Providence, RI, 1991, pp. 215–254. Translated from the Russian by V. E. Nazaĭkinskiĭ. MR**1118164**

## Bibliographic Information

**Artur Avila**- Affiliation: CNRS, IMJ-PRG, UMR 7586, Univ Paris Diderot, Sorbonne Paris Cité, Sorbonnes Universités, UPMC Univ Paris 06, F-75013, Paris, France; IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil
- Email: artur@math.univ-paris-diderot.fr
- Received by editor(s): October 11, 2012
- Received by editor(s) in revised form: April 11, 2014
- Published electronically: June 11, 2014
- © Copyright 2014 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**28**(2015), 579-616 - MSC (2010): Primary 37H15; Secondary 47B39
- DOI: https://doi.org/10.1090/S0894-0347-2014-00814-6
- MathSciNet review: 3300702