Large-coupling absence of ground states in the infrared-critical spin boson model

Let $H(\lambda,0)$ denote the spin boson Hamiltonian with coupling constant $\lambda$ and zero magnetic field. Assume that the interaction function satisfies

$$v\in\mathcal{D}(\omega^{-1/2})\setminus\mathcal{D}(\omega^{-1}).$$

Large-coupling absence conjecture. There exists $\lambda_{\mathsf c}>0$ such that $H(\lambda,0)$ has no ground state for

$$|\lambda|>\lambda_{\mathsf c}.$$

The conjecture concerns the open large-coupling regime of the infrared-critical spin boson model without magnetic field. Heuristic evidence comes from its connection with a long-range one-dimensional Ising model whose interaction decays as $1/t^2$; the rigorous treatment of this regime remains open.