Infrared Problem for the Nelson Model on Static Space-Times

We consider the Nelson model on some static space-times and investigate the problem of existence of a ground state. Nelson models with variable coefficients arise when one replaces in the usual Nelson model the flat Minkowski metric by a static metric, allowing also the boson mass to depend on position. We investigate the existence of a ground state of the Hamiltonian in the presence of the infrared problem, i. e. assuming that the boson mass m(x) tends to 0 at spatial infinity. We show that if m(x) ≥ C {pipe}x{pipe}^-1 at infinity for some C > 0 then the Nelson Hamiltonian has a ground state.