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 ∆ = −∇² + M² on Euclideanized AdS₃ with the Euclideanized time coordinate compactified with period β, H₃/Z, by explicitly computing its eigenvalues and computing their product. Upon assuming that eigenfunctions are normalizable on H₃/Z, we found that there are no such eigenfunctions. Upon closer examination, we discover that the intuition that H₃/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 $\widetilde{\text{∆}}$ \( \overset{\sim }{\Delta } \) = r²∆, which has the property that (1) the determinant of ∆ and the determinant of r²∆ have the same dependence on β, and that (2) the Green’s function of ∆ can be spectrally decomposed into eigenfunctions of $\widetilde{\text{∆}}$ \( \overset{\sim }{\Delta } \) . We identify the $\widetilde{\text{∆}}$ \( \overset{\sim }{\Delta } \) operator as the “weighted Laplacian” in the context of warped compactifications, and comment on possible applications.