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On the inverse spectral problem for the quasi-periodic Schrödinger equation
David Damanik1 ; Michael Goldstein2
1 Department of Mathematics, Rice University 6100 S. Main St. 77005-1892 Houston TX USA
2 Department of Mathematics, University of Toronto Bahen Centre, 40 St. George St. M5S 2E4 Toronto Ontario Canada
Publications Mathématiques de l'IHÉS, Volume 119 (2014), pp. 217-401
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We study the quasi-periodic Schrödinger equation

-ψ '' (x)+V(x)ψ(x)=Eψ(x),x∈𝐑
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|).

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DOI: 10.1007/s10240-013-0058-x
Keywords: Implicit Function Theorem, Inductive Assumption, Simple Eigenvalue, Principal Point, Inverse Spectral Problem

David Damanik 1; Michael Goldstein 2

1 Department of Mathematics, Rice University 6100 S. Main St. 77005-1892 Houston TX USA
2 Department of Mathematics, University of Toronto Bahen Centre, 40 St. George St. M5S 2E4 Toronto Ontario Canada 
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     title = {On the inverse spectral problem for the quasi-periodic {Schr\"odinger} equation},
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David Damanik; Michael Goldstein. On the inverse spectral problem for the quasi-periodic Schrödinger equation. Publications Mathématiques de l'IHÉS, Volume 119 (2014), pp. 217-401. doi: 10.1007/s10240-013-0058-x

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