Stability of discrete ground state

Abstract

We present new criteria for a self-adjoint operator to have a ground state. 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.

Existence of a ground state for the Nelson model with a singular perturbation

“…In this model, the dispersion relation is supposed to be strictly positive and the Hamiltonian is defined on a boson Fock space. Miyao and Sasaki [20] show the existence of the ground state for a generalized spinboson model with φ 2 -perturbation, and it is not supposed that the particle Hamiltonian has a compact resolvent. Takaesu [24] shows the existence of a ground state for a generalized spin-boson model with a singular perturbation of the form (1.3) but for sufficiently small coupling constants.…”

Self-adjointness of the generalized spin-boson Hamiltonian with a quadratic boson interaction

“…In the following arguments, we show that each term of ( 8)-( 10) is greater than or equal to −C( A 1/2 Ψ 2 + I ⊗ (dΓ(W ) + 1) 1/2 Ψ 2 ) for some positive constant C. First, we estimate the term (8). Since [A 1/2 , B j ] is bounded,…”

“…This Hamiltonian H(λ, µ) was studied by Miyao and Sasaki [8]. In the case of µ = 0, it was introduced in [1] and called the generalized spin-boson (abbreviated as GSB) Hamiltonian.…”

“…Arai and Hirkawa [1] introduced an abstract non-relativistic quantum field model which is a generalization of the spin-boson model and it is called the generalized spin-boson model. Miyao and Sasaki [8] added φ 2 term to the generalized spin-boson Hamiltonian. They showed that the Hamiltonian is self-adjoint for small coupling constants by applying the Kato-Rellich theorem.…”

Ground states of the massless Dereziński–Gérard model

“…The existence of a ground state of the massive Dereziński-Gérard model was investigated by Dereziński and Gérard 5 and Miyao and Sasaki. 10 Dereziński and Gérard 5 showed the existence of a ground state of the massive Dereziński-Gérard model by specifying the essential spectrum of the Hamiltonian, which is an analog of the HVZ theorem in many body problems. Miyao and Sasaki 10 showed the existence of a ground state of the massive Dereziński-Gérard model by applying the method developed by Arai and Hirokawa.…”