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

Abstract

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.

Keywords

Hyperbolic surfaces / Spectral theory / Twisted Laplacian / 58J51 / 58J50 / 35P15

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Funding

National Key R&D Program of China(2022YFA100740)

RIGHTS & PERMISSIONS

Peking University