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

Christian Gérard (1) , Fumio Hiroshima (2) , Annalisa Panati (3, 4) , A. Suzuki (5)

1 LM-Orsay - Laboratoire de Mathématiques d'Orsay
2 Department of Mathematics
3 CPT - Centre de Physique Théorique - UMR 6207
4 CPT - Centre de Physique Théorique - UMR 7332
5 Departement of Materials Science and Engineering

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 $0$ at spatial infinity. Using path space techniques, we show that if $m(x)\leq C |x|^{-\mu}$ at infinity for some $C>0$ and $\mu>1$ then the Nelson Hamiltonian has no ground state.

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Christian Gérard, Fumio Hiroshima, Annalisa Panati, A. Suzuki. Absence of ground state for the Nelson model on static space-times. Journal of Functional Analysis, 2012, 262 (01), pp.273-299. ⟨10.1016/j.jfa.2011.09.010⟩. ⟨hal-00544552⟩