Sufficiency of τ_{α(λ),1}-integrability for the existence criterion
Ascertain whether integrability of the one-point two-time correlation function τ_{α(λ),1}(t) of the associated continuum Ising model (i.e., ∫_0^∞ τ_{α(λ),1}(t) dt < ∞) implies the uniform energy-derivative bound limsup_{T→∞} (1/T) ∫_0^T ∫_0^T E_{α(λ),T}[X_t X_s] dt ds < ∞ required for ground-state existence of the spin boson Hamiltonian, thereby allowing the criterion to be stated purely in terms of τ_{α(λ),1}.
References
It would be desirable to phrase the above integrability criterion in terms of $\tau_{\alpha(\lambda),1}$ instead of more general correlation functions. Using monotonicicty of correlations in the domain, it is not difficult to see that the integrability of $t\mapsto \tau_{\alpha(\lambda),1}(t)$ is necessary for \cref{Equation: Condition for existence} to hold. However, it is unclear to the authors if the reverse implication holds.