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

A cuspidal end is a type of metric singularity, described as a product $S^1\times ]a,+\mathrm{\infty }[$ with the Poincaré metric. The underlying set can also be seen as $\mathbb{R}\times ]a,+\mathrm{\infty }[$ subject to the action of the translation $T:(x,y)⟶(x+1,y)$. 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 ${\mathrm{\Delta }}_{L,0}$ 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 $\det ({\mathrm{\Delta }}_{L,0}+\mu )$, as $\mu >0$ goes to infinity for any $a$, and also as $a$ goes to infinity for $\mu =0$.