On the inverse spectral problem for the quasi-periodic Schrödinger equation

Damanik, David; Goldstein, Michael

Publications Mathématiques de l'IHÉS, Tome 119 (2014), pp. 217-401

We study the quasi-periodic Schrödinger equation

$$-{\psi }^{\text{'}\text{'}}\left(x\right)+V\left(x\right)\psi \left(x\right)=E\psi \left(x\right),x\in 𝐑$$

in the regime of “small” $V$. Let $(E_m^{\text{'}},E_m^{\text{'}\text{'}})$, $m\in {𝐙}^{\nu }$, be the standard labeled gaps in the spectrum. Our main result says that if ∈ $E_m^{\text{'}\text{'}}-E_m^{\text{'}}\le \epsilon exp(-{\kappa }_0\left|m\right|)$ for all $m\in {𝐙}^{\nu }$, with $\epsilon$ being small enough, depending on κ₀>0 and the frequency vector involved, then the Fourier coefficients of $V$ obey $\left|c\left(m\right)\right|\le {\epsilon }^{1/2}exp(-\frac{{\kappa }_0}{2}\left|m\right|)$ for all $m\in {𝐙}^{\nu }$. On the other hand we prove that if |c(m)|≤εexp(−κ₀|m|) with $\epsilon$ being small enough, depending on κ₀>0 and the frequency vector involved, then $E_m^{\text{'}\text{'}}-E_m^{\text{'}}\le 2\epsilon exp(-\frac{{\kappa }_0}{2}\left|m\right|)$.

Keywords: Implicit Function Theorem, Inductive Assumption, Simple Eigenvalue, Principal Point, Inverse Spectral Problem

[Av1] A. Avila, Global theory of one-frequency Schrödinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity, preprint (2009), | arXiv

[Av2] A. Avila, Global theory of one-frequency Schrödinger operators II: acriticality and finiteness of phase transitions for typical potentials, preprint. | MR

[Av3] A. Avila, Almost reducibility and absolute continuity I, preprint (2010), | arXiv

[Bo] Bourgain, J. Green’s Function Estimates for Lattice Schrödinger Operators and Applications (2004) | MR | Zbl

[BoJi] Bourgain, J.; Jitomirskaya, S. Absolutely continuous spectrum for 1D quasi-periodic operators, Invent. Math., Volume 148 (2002), pp. 453-463 | MR | Zbl | DOI

[DiSi] Dinaburg, E. I.; Sinai, Y. G. The one dimensional Schrödinger equation with quasiperiodic potential, Funkc. Anal. Prilozh., Volume 9 (1975), pp. 8-21 | Zbl

[DG] D. Damanik and M. Goldstein, On the existence and uniqueness of global solutions of the KdV equation with quasi-periodic initial data, preprint (2012), | arXiv

[El] Eliasson, H. Floquet solutions for the 1-dimensional quasiperiodic Schrödinger equation, Commun. Math. Phys., Volume 146 (1992), pp. 447-482 | Zbl | DOI

[EK] Eliasson, H.; Kuksin, S. KAM for the non-linear Schrödinger equation, Ann. Math., Volume 172 (2010), pp. 371-435 | Zbl | DOI

[FrSp] Fröhlich, J.; Spencer, T. Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys., Volume 88 (1983), pp. 151-189 | Zbl | DOI

[FSW] Fröhlich, J.; Spencer, T.; Wittwer, P. Localization for a class of one dimensional quasi-periodic Schrödinger operators, Commun. Math. Phys., Volume 132 (1990), pp. 5-25 | Zbl | DOI

[HA] Hadj Amor, S. Hölder continuity of the rotation number for quasi-periodic co-cycles in SL(2,R), Commun. Math. Phys., Volume 287 (2009), pp. 565-588 | Zbl | DOI

[Lax] Lax, P. Integrals of non-linear equations of evolution and solitary waves, Commun. Pure Appl. Math., Volume 21 (1968), pp. 467-490 | Zbl | DOI