Sufficiency of two-point function integrability for the existence criterion

Determine whether integrability of the one-sided continuum Ising two-point function t ↦ τ_{α(λ),1}(t), i.e., ∫_0^∞ τ_{α(λ),1}(t) dt < ∞, is sufficient to imply the existence criterion limsup_{T→∞} (1/T) ∫_0^T ∫_0^T E_{α(λ),T}[X_t X_s] dt ds < ∞ used to guarantee a ground state of the spin boson Hamiltonian H_λ, where E_{α(λ),T} denotes expectation under the continuum Ising model with kernel g(t)=∫_{R^d} |v(k)|^2 e^{-|t| ω(k)} dk and coupling α(λ)=λ^2/8.

Background

Proposition prop:existencecorbound reduces existence of a spin–boson ground state to a uniform-in-time integrability condition involving the full two-time correlation function E_{α,T}[X_t X_s]. The authors note that integrability of the one-dimensional two-point function τ_{α,1} is necessary for this criterion.

However, it is not known whether the converse implication holds, i.e., whether the simpler condition on τ_{α,1} alone suffices to ensure the more global integrability condition required by the proposition. Clarifying this would streamline the existence criterion and potentially simplify applications.