Pseudo-Laplacian on a cuspidal end with a flat unitary line bundle: Alvarez--Wentworth boundary conditions

TL;DR

This paper investigates the asymptotic behavior of the zeta-regularized determinant of a pseudo-Laplacian on a cuspidal end with a flat line bundle, under Alvarez--Wentworth boundary conditions, as parameters grow large.

Contribution

It provides the first analysis of the determinant's asymptotics for pseudo-Laplacians with Alvarez--Wentworth boundary conditions on cuspidal ends.

Findings

Asymptotic behavior of the determinant as $\mu o \infty$ for fixed $a$.

Asymptotic behavior of the determinant as $a o \infty$ for $\mu=0$.

Extension of spectral analysis to cuspidal ends with flat line bundles.

Abstract

A cuspidal end is a type of metric singularity, described as a product $S^1 \times \left] a, +\infty \right[$ with the Poincar\'e metric. The underlying set can also be seen as $\mathbb{R} \times \left] a, +\infty \right[$ subject to the action of the translation $T : \left( x,y \right) \longrightarrow \left( x+1, y \right)$. On it, one may consider a holomorphic line bundle $L$, coming from a unitary character of the group generated by $T$. The complex modulus induces a flat metric on $L$, and a pseudo-Laplacian $\Delta_{L,0}$ acting on functions can be associated to the Chern connection. One needs to specify boundary conditions, and they are here chosen to be the Alvarez--Wentworth boundary conditions, which are a combination of Dirichlet and Neumann boundary conditions. The aim of this paper is to find the asymptotic behavior of the zeta-regularized determinant $\det \left( \Delta_{L,0} + \mu \right)$, as $\mu > 0$ goes to infinity for any $a$, and also as $a$ goes to infinity for $\mu = 0$.