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<p>We investigate the asymptotic spectral distribution of the twisted Laplacian associated with a real harmonic 1-form on a compact hyperbolic surface. In particular, we establish a sublinear lower bound on the number of eigenvalues in a sufficiently large strip determined by the pressure of the harmonic 1-form. Furthermore, following an observation by Anantharaman [Geom. Funct. Anal. <b>20</b>(3), 593–626 (2010)], we show that quantum unique ergodicity fails to hold for certain twisted Laplacians.</p>

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Sublinear Lower Bounds of Eigenvalues for Twisted Laplacian on Compact Hyperbolic Surfaces

  • 期刊论文

  • 发表日期：

    2026年3月31日

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  • Yulin Gong,
  • Long Jin

摘要

We investigate the asymptotic spectral distribution of the twisted Laplacian associated with a real harmonic 1-form on a compact hyperbolic surface. In particular, we establish a sublinear lower bound on the number of eigenvalues in a sufficiently large strip determined by the pressure of the harmonic 1-form. Furthermore, following an observation by Anantharaman [Geom. Funct. Anal. 20(3), 593–626 (2010)], we show that quantum unique ergodicity fails to hold for certain twisted Laplacians.