The Isospectral Torus of Quasi-Periodic Schr\"odinger Operators via Periodic Approximations

TL;DR

This paper characterizes the set of reflectionless quasi-periodic Schrödinger operators as a torus and describes their potential functions via periodic approximations, extending multi-scale analysis techniques.

Contribution

It introduces a new approach to analyze isospectral reflectionless potentials using periodic approximations and multi-scale analysis, providing stronger estimates than previous Hill operator theories.

Findings

The set of reflectionless potentials is homeomorphic to a torus.

Reflectionless potentials share the same frequency vector   with the original potential.

Fourier coefficients of isospectral potentials are tightly bounded.

Abstract

We study the quasi-periodic Schr\"odinger operator $$ -\psi"(x) + V(x) \psi(x) = E \psi(x), \qquad x \in \mathbb{R} $$ in the regime of "small" $V(x) = \sum_{m\in\mathbb{Z}^\nu}c(m)\exp (2\pi i m\omega x)$, $\omega = (\omega_1, \dots, \omega_\nu) \in \mathbb{R}^\nu$, $|c(m)| \le \varepsilon \exp(-\kappa_0|m|)$. 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 $Q (x) = \sum_{m \in \mathbb{Z}^\nu} d(m) \exp (2\pi i m\omega x)$, with the same $\omega$ and with $|d(m)| \le \sqrt{2 \varepsilon} \exp(-\frac{\kappa_0}{2} |m|)$. Our derivation relies on the study of the approximation via Hill operators with potentials $\tilde V (x) = \sum_{m \in \mathbb{Z}^\nu} c(m) \exp (2 \pi i m \tilde \omega x)$, where $\tilde \omega$ is a rational approximation of $\omega$. 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 $|\omega - \tilde \omega| \thicksim \tilde T^{-\delta}$, $0 < \delta < 1/2$ is slow, on the scale of the period $\tilde T$ of the Hill operator.