Absence of ground state for the Nelson model on static space-times

We consider the Nelson model on some static space times and investigate the problem of absence 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 absence 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 Oat spatial infinity. Using path space techniques, we show that if m(x) <= C vertical bar x vertical bar(-mu) at infinity for some C > 0 and mu > I then the Nelson Hamiltonian has no ground state. (C) 2011 Elsevier Inc. All rights reserved.