Path integral of free fields and the determinant of Laplacian in warped space-time

We revisit the problem of computing the determinant of Klein-Gordon operator $\mathrm{\Delta }=-{\mathrm{\nabla }}^2+M^2$ on Euclideanized $Ad{S}_3$ with the Euclideanized time coordinate compactified with period $\beta$, $H_3\mathrm{/}Z$, by explicitly computing its eigenvalues and computing their product. Upon assuming that eigenfunctions are normalizable on $H_3\mathrm{/}Z$, we found that there are no such eigenfunctions. Upon closer examination, we discover that the intuition that $H_3\mathrm{/}Z$ is like a box with normalizable eigenfunctions was false, and that there is, instead, a set of eigenfunctions which forms a continuum. Somewhat to our surprise, we find that there is a different operator $\tilde{\mathrm{\Delta }}=r^2\mathrm{\Delta }$, which has the property that (1) the determinant of $\mathrm{\Delta }$ and the determinant of $r^2\mathrm{\Delta }$ have the same dependence on $\beta$, and that (2) the Green’s function of $\mathrm{\Delta }$ can be spectrally decomposed into eigenfunctions of $\tilde{\mathrm{\Delta }}$. We identify the $\tilde{\mathrm{\Delta }}$ operator as the ``weighted Laplacian’’ in the context of warped compactifications, and comment on possible applications.