Stability of discrete ground state

Abstract

We present new criteria for a self-adjoint operator to have a ground state. Although the proof which we give is easy, it has many applications. And in various models, it is easy to check the criteria. As an application, we consider models of "quantum particles" coupled to a massive Bose field and prove the existence of a ground state of them, where the particle Hamiltonian does not necessarily have compact resolvent.