arXiv 2602.074452026-02math.SP

多频准周期Schrödinger算子谱区间化的普遍性

On the Genericity of the Spectrum Intervalization for Multi-Frequency Quasiperiodic Schrödinger Operators

Ocean University of China（中国海洋大学）

Daxiong Piao

备注：: This manuscript is withdrawn due to substantial mathematical errors in the formulation and proof of the main theorem. In particular, the key quantitative estimates used in the argument do not hold as stated. Consequently, the claimed generic interval-spectrum result has not been established

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AI总结：

本文证明了Goldstein等人关于多频准周期Schrödinger算子谱区间化的普遍性猜想，表明在强耦合条件下，几乎所有实三角多项式势系数的谱形成单一区间。

摘要：

本文证明了Goldstein、Schlag和Voda[Invent. Math. 217(2019)]关于多频准周期Schrödinger算子谱区间化的普遍性猜想。具体来说，我们证明在强耦合条件下，几乎所有实三角多项式势系数的谱形成单一区间。这证实了Chulaevky和Sinai[Comm.Math.Phys. 125(1989)]长期以来的直觉，即对于通用势，谱通常为区间，并将Goldstein等人的存在结果扩展到全测度设置。我们的证明依赖于微分拓扑、测度论和解析函数论的工具。

英文摘要：

This paper proves a genericity conjecture by Goldstein, Schlag, and Voda[Invent. Math.\textbf{217}(2019)] for multi-frequency quasiperiodic Schrödinger operators. Specifically, we show that for almost all coefficients of real trigonometric polynomial potentials, the spectrum forms a single interval under strong coupling conditions. This confirms a long-standing intuition by Chulaevky and Sinai[Comm.Math.Phys.\textbf{125}(1989)] that the spectrum typically intervals for generic potentials, and extends the existence results of Goldstein et al. to a full measure setting. Our proof relies on tools from differential topology, measure theory, and analytic function theory.

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