Spectrum of the Laplacian on the Page metric

Abstract

We numerically construct the spectrum of the Laplacian on Page's inhomogeneous Einstein metric on ${\mathbb{CP}}^2\#{\overline{\mathbb{CP}}}^2$ by reducing the problem to a (singular) Sturm-Liouville problem in one dimension. We perform a perturbative analysis based upon a closely related, exactly solvable problem that strongly supports our results. We also study the spectrum of the Lichnerowicz Laplacian on symmetric traceless transverse two-tensors. The method relies on both the isometries of the Page metric and pseudospectral methods to numerically solve the resulting ODEs.