On the Inverse Spectral Problem for the Quasi-Periodic Schrödinger Equation

We study the quasi-periodic Schrödinger equation $$ -ψ"(x) + V(x) ψ(x) = E ψ(x), \qquad x \in \IR $$ 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 $E"_m - E'_m \le \ve \exp(-κ_0 |m|)$ for all $m \in \zv$, with $\ve$ being small enough, depending on $κ_0 > 0$ and the frequency vector involved, then the Fourier coefficients of $V$ obey $|c(m)| \le \ve^{1/2} \exp(-\frac{κ_0}{2} |m|)$ for all $m \in \zv$. On the other hand we prove that if $|c(m)| \le \ve \exp(-κ_0 |m|)$ with $\ve$ being small enough, depending on $κ_0 > 0$ and the frequency vector involved, then $E"_m - E'_m \le 2 \ve \exp(-\frac{κ_0}{2} |m|)$.

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