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

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