The isospectral torus of quasi-periodic Schrödinger operators via periodic approximations

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• 作者：David Damanik ; Michael Goldstein ; Milivoje Lukic
• 刊名：Inventiones mathematicae
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：207
• 期：2
• 页码：895-980
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer Berlin Heidelberg
• ISSN：1432-1297
• 卷排序：207

文摘

We study the quasi-periodic Schrödinger operator $$\begin{aligned} -\psi ''(x) + V(x) \psi (x) = E \psi (x), \quad x \in {\mathbb {R}}\end{aligned}$$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 Damanik and Goldstein (Publ Math Inst Hautes Études Sci 119:217–401, 2014) applies to these Hill operators. Namely, in Damanik et al. (Trans Am Math Soc, to appear, arXiv:1409.2147, 2016) we developed the multi-scale analysis for the operators dual to the Hill operators in question. The main estimates obtained in Damanik et al. (Trans Am Math Soc, to appear, arXiv:1409.2147, 2016) allow us here to establish estimates for the gap lengths and the Fourier coefficients in a form that 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.D. D. was partially supported by NSF Grants DMS-1067988 and DMS-1361625. M. G. was partially supported by NSERC. M. G. expresses his gratitude for the hospitality during a stay at the Institute of Mathematics at the University of Stony Brook in May 2014. M. L. was partially supported by NSF Grant DMS-1301582.