On the inverse spectral problem for the quasi-periodic Schrödinger equation
Publications Mathématiques de l'IHÉS, Tome 119 (2014), pp. 217-401.

We study the quasi-periodic Schrödinger equation

 $-{\psi }^{\text{'}\text{'}}\left(x\right)+V\left(x\right)\psi \left(x\right)=E\psi \left(x\right),\phantom{\rule{1em}{0ex}}x\in 𝐑$
in the regime of “small” $V$. Let $\left({E}_{m}^{\text{'}},{E}_{m}^{\text{'}\text{'}}\right)$, $m\in {𝐙}^{\nu }$, be the standard labeled gaps in the spectrum. Our main result says that if ∈ ${E}_{m}^{\text{'}\text{'}}-{E}_{m}^{\text{'}}\le \epsilon exp\left(-{\kappa }_{0}|m|\right)$ 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 $|c\left(m\right)|\le {\epsilon }^{1/2}exp\left(-\frac{{\kappa }_{0}}{2}|m|\right)$ 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 ${E}_{m}^{\text{'}\text{'}}-{E}_{m}^{\text{'}}\le 2\epsilon exp\left(-\frac{{\kappa }_{0}}{2}|m|\right)$.

DOI : https://doi.org/10.1007/s10240-013-0058-x
MANUSCRIPT : 58
PUBLISHER-ID : s10240-013-0058-x
Mots clés : Implicit Function Theorem, Inductive Assumption, Simple Eigenvalue, Principal Point, Inverse Spectral Problem
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author = {Damanik, David and Goldstein, Michael},
title = {On the inverse spectral problem for the quasi-periodic {Schr\"odinger} equation},
journal = {Publications Math\'ematiques de l'IH\'ES},
pages = {217--401},
publisher = {Springer Berlin Heidelberg},
volume = {119},
year = {2014},
doi = {10.1007/s10240-013-0058-x},
zbl = {1296.35168},
mrnumber = {3210179},
language = {en},
url = {http://archive.numdam.org/articles/10.1007/s10240-013-0058-x/}
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Damanik, David; Goldstein, Michael. On the inverse spectral problem for the quasi-periodic Schrödinger equation. Publications Mathématiques de l'IHÉS, Tome 119 (2014), pp. 217-401. doi : 10.1007/s10240-013-0058-x. http://archive.numdam.org/articles/10.1007/s10240-013-0058-x/

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