Spectral properties of monopoles and gravitational instantons

Abstract

Motivated by spectral problems arising in gauge theory and gravity, we develop a rigorous proof of infinitely many bound states for a family of radial Laplacians that have Calogero characteristics at the origin and Coulombic ones at infinity. We first study the spectrum of the operator obtained by linearising the Yang-Mills-Higgs equations around a charge one monopole. We then study two Laplace operators on four-dimensional Riemannian manifolds, namely the Laplace operator on the Atiyah-Hitchin moduli space of centred charge two monopoles and the Laplace operator associated with the Taub-bolt family. We apply our theorem to each operator proving they have infinite discrete spectrums and numerically compute the eigenvalues. We compare results to appropriate analytic approximations for each case.