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<p>For quasi-periodic Schrödinger operators with small Gasymov-type potentials and arbitrary irrational frequencies, we establish a complete spectral characterization: the spectrum coincides with that of the discrete free Laplacian. Our result is non-perturbative in the sense that the smallness condition is independent of the frequency. Furthermore, we prove the absence of both residual and point spectra, thereby establishing purely continuous spectrum. The proof combines Avila’s global theory, non-perturbative almost reducibility, and Green’s function estimates.</p>

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Non-perturbative Isospectrality for Quasi-periodic Gasymov-Type Potentials

  • 期刊论文

  • 发表日期：

    2026年4月4日

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  • Jiawei He,
  • Xueyin Wang,
  • Zhenfu Wang

摘要

For quasi-periodic Schrödinger operators with small Gasymov-type potentials and arbitrary irrational frequencies, we establish a complete spectral characterization: the spectrum coincides with that of the discrete free Laplacian. Our result is non-perturbative in the sense that the smallness condition is independent of the frequency. Furthermore, we prove the absence of both residual and point spectra, thereby establishing purely continuous spectrum. The proof combines Avila’s global theory, non-perturbative almost reducibility, and Green’s function estimates.