On the Inverse Spectral Problem for the Quasi-Periodic Schr\"odinger Equation

We study the quasi-periodic Schr"odinger equation $$-\psi \mathrm{"}(x)+V(x)\psi (x)=E\psi (x),x\in \text{IR}$$ in the regime of “small” $V$. Let $(E_m^{\mathrm{\prime }},E{\mathrm{"}}_m)$, $m\in \text{zv}$, be the standard labeled gaps in the spectrum. Our main result says that if $E{\mathrm{"}}_m-E_m^{\mathrm{\prime }}\le \text{ve}\exp (-{\kappa }_0\mathrm{\mid }m\mathrm{\mid })$ for all $m \in \zv$,with$\ve$beingsmallenough,dependingon$\kappa_0 > 0$ and the frequency vector involved, then the Fourier coefficients of $V$ obey $|c(m)| \le \ve^{1/2} \exp(-\frac{\kappa_0}{2} |m|)$forall$m \in \zv$. On the other hand we prove that if $\mathrm{\mid }c(m)\mathrm{\mid }\le \text{ve}\exp (-{\kappa }_0\mathrm{\mid }m\mathrm{\mid })$ with $\text{ve}$ being small enough, depending on ${\kappa }_0>0$ and the frequency vector involved, then $E{\mathrm{"}}_m-E_m^{\mathrm{\prime }}\le 2\text{ve}\exp (-\frac{{\kappa }_0}{2}\mathrm{\mid }m\mathrm{\mid })$.