On the spectrum of twisted Laplacians and the Teichm\"uller representation

TL;DR

This paper studies the spectral properties of non self-adjoint Laplacians with non unitary twists on hyperbolic surfaces, revealing how their spectra relate to Teichm"uller deformations and providing refined eigenvalue counting estimates.

Contribution

It offers a detailed analysis of the spectrum of twisted Laplacians for Teichm"uller type representations, including high energy limits and eigenvalue counting improvements.

Findings

Spectrum lies inside a parabola in the complex plane.

Weyl's law holds in the bulk of the spectrum, determined by critical exponents.

Provides polynomial improvement over classical eigenvalue counting estimates.

Abstract

We consider Laplacians with non unitary twists acting on sections of flat vector bundles over compact hyperbolic surfaces. These non self-adjoint Laplacians have discrete spectrum inside a parabola in the complex plane. For representations of the fundamental group of the base surface which are of Teichm\"uller type, we investigate the high energy limit and give a precise description of the bulk of the spectrum where Weyl's law is satisfied in terms of critical exponents of the representations which are completely determined by the Manhattan curve associated to the Teichm\"uller deformation. Our main result provides a counting estimate for the eigenvalues outside the bulk with a polynomial improvement over Weyl's law.