The Isospectral Torus of Quasi-Periodic Schrödinger Operators via Periodic Approximations

We study the quasi-periodic Schrödinger operator $$ -ψ"(x) + V(x) ψ(x) = E ψ(x), \qquad x \in \mathbb{R} $$ in the regime of "small" $V(x) = \sum_{m\in\mathbb{Z}^ν}c(m)\exp (2πi mωx)$, $ω= (ω_1, \dots, ω_ν) \in \mathbb{R}^ν$, $|c(m)| \le \varepsilon \exp(-κ_0|m|)$. We show that the set of reflectionless potentials isospectral with $V$ is homeomorphic to a torus. Moreover, we prove that any reflectionless potential $Q$ isospectral with $V$ has the form $Q (x) = \sum_{m \in \mathbb{Z}^ν} d(m) \exp (2πi mωx)$, with the same $ω$ and with $|d(m)| \le \sqrt{2 \varepsilon} \exp(-\frac{κ_0}{2} |m|)$. Our derivation relies on the study of the approximation via Hill operators with potentials $\tilde V (x) = \sum_{m \in \mathbb{Z}^ν} c(m) \exp (2 πi m \tilde ωx)$, where $\tilde ω$ is a rational approximation of $ω$. It turns out that the multi-scale analysis method of \cite{DG} applies to these Hill operators. Namely, in \cite{DGL} we developed the multi-scale analysis for the operators dual to the Hill operators in question. The main estimates obtained in \cite{DGL} allow us here to establish the estimates for the gap lengths and the Fourier coefficients in a form which is considerably stronger than the estimates known in the theory of Hill operators with analytic potentials in the general setting. Due to these estimates, the approximation procedure for the quasi-periodic potentials is effective, despite the fact that the rate of approximation $|ω- \tilde ω| \thicksim \tilde T^{-δ}$, $0 < δ< 1/2$ is slow, on the scale of the period $\tilde T$ of the Hill operator.

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