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

-ψ '' (x)+V(x)ψ(x)=Eψ(x),x𝐑
in the regime of “small” V . Let (E m ' ,E m '' ), m 𝐙 ν , be the standard labeled gaps in the spectrum. Our main result says that if ∈ E m '' -E m ' εexp(-κ 0 |m|) for all m 𝐙 ν , with ε being small enough, depending on κ0>0 and the frequency vector involved, then the Fourier coefficients of V obey |c(m)|ε 1/2 exp(-κ 0 2|m|) for all m 𝐙 ν . On the other hand we prove that if |c(m)|≤εexp(−κ0|m|) with ε being small enough, depending on κ0>0 and the frequency vector involved, then E m '' -E m ' 2εexp(-κ 0 2|m|).

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},
     address = {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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