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 $E_m^{\text{'}\text{'}}-E_m^{\text{'}}\le 2\epsilon exp(-\frac{{\kappa }_0}{2}\left|m\right|)$.

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