Neumann scalar determinants on constant curvature disks

TL;DR

This paper derives infinite series representations for the logarithms of massive scalar determinants with Neumann boundary conditions on constant curvature disks, extending previous Dirichlet determinant results and providing exact formulas for specific mass values.

Contribution

It introduces new series representations for Neumann scalar determinants on constant curvature disks, relating them to Dirichlet determinants via boundary maps, and extends prior Dirichlet results.

Findings

Derived series representations for Neumann determinants.

Established relations between Neumann and Dirichlet determinants.

Obtained exact formulas for specific mass values.

Abstract

Working in the $\zeta$-function regularisation scheme, we find certain infinite series representations of the logarithms of massive scalar determinants, $\det(\Delta+m^{2})$ for arbitrary $m^2$, on finite round disks of constant curvature ($R=\frac{2\eta}{L^2}, \eta=0,\pm1$) with Neumann boundary conditions. The derivation of these representations relies on a relation between the Neumann determinants on the disks and the corresponding Dirichlet determinants via the determinants of the Dirichlet-to-Neumann maps on the boundaries of the disks. We corroborate the results in an appendix by computing the Neumann determinants in an alternative way. In the cases of disks with nonzero curvatures, we show that the infinite series representations reduce to exact expressions for some specific values of $m^2$, viz. $m^2=-\frac{\eta}{L^2}q(q+1)$ with $q\in \mathbb{N}$. Our analysis uses and extends the results obtained in arXiv:2405.14958 for similar Dirichlet determinants on constant curvature disks.