Sublinear lower bounds of eigenvalues for twisted Laplacian on compact hyperbolic surfaces
Yulin Gong, Long Jin
TL;DR
This work studies the spectrum of the twisted Laplacian Δ_ω on a compact hyperbolic surface X with a real harmonic 1-form ω. By combining the twisted Selberg trace formula with thermodynamic formalism and large-deviation techniques, the authors establish a sublinear lower bound for the spectral count N_A(R) in strips governed by the pressure Pr(ω) and the stable norm ‖ω‖_s, and they derive explicit lower bounds for the essential spectral gap G_ω, including G_ω≥2‖ω‖_s−Pr(ω)−1/2 and G_ω≥(3/2)Pr(2ω)−2Pr(ω)−1/2, with refinements in the arithmetic case. Building on Anantharaman’s observations, they show that a positive essential gap implies the failure of quantum unique ergodicity for Δ_ω by comparing distinct semiclassical defect measures. The paper also clarifies the role of the pressure and stable norm in shaping spectral and dynamical properties, and provides an alternative arithmetic-proof route to ess-gap bounds, linking to twisted Selberg zeta functions Z_ω(s). The results illuminate how nonunitary twists perturb the high-frequency spectral distribution and connect spectral gaps to dynamical quantities of the geodesic flow.
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 \cite{nalinideviation}, we show that quantum unique ergodicity fails to hold for certain twisted Laplacians.