Binding condition for a general class of quantum field Hamiltonians

Abstract

We consider a system of a quantum particle interacting with a quantum field and an external potential $V(\bx)$. The Hamiltonian is defined by a quadratic form $H^V = H^0 + V(\bx)$, where $H^0$ is a quadratic form which preserves the total momentum. $H^0$ and $H^V$ are assumed to be bounded from below. We give a criterion for the positivity of the binding energy $E_\mathrm{bin} = E^0-E^V$, where $E^0$ and $E^V$ are the ground state energies of $H^0$ and $H^V$. As examples of the result, the positivity of the binding energy of the semi-relativistic Pauli-Fierz model and Nelson type Hamiltonian is proved.