A new approach to continuous multi-scale analysis in nonrelativistic QED: ground states and photon number bounds for the spin-boson model with critical infrared singularity

Abstract

We introduce a new method for the study of spectral problems in nonrelativistic quantum electrodynamics, NR QED. The method is based on the so-called Pizzo’s Method (Ann Henri Poincare 4(3):439–486, 2003) or multi-scale analysis. We use it to prove existence and uniqueness of ground-state eigenvalues for the spin-boson model with critical infrared behavior. Denoting by H the corresponding Hamiltonian, we construct its ground-state eigenvalue using a continuous family of infrared cutoff Hamiltonians \((H_t)_{t \ge 0}\) such that as the parameter t tends to infinity, the cutoff is removed. The continuous method we introduce in this paper enjoys many advantages (and it has no disadvantages) with respect to the discrete version (where the parameter t takes its values in a discrete set). The reason is that it is a natural extension of the discrete iteration method, and therefore, it incorporates at the same time both: tools from the discrete approach and techniques from the area of differential equations. Actually, in our proof the analysis of the cutoff Hamiltonians \((H_t)_{t \in [0, \infty )}\) is carried out with the help of our calculations and proofs in Bach et al. (J Math Anal Appl 453:773–797, 2017). However, our main argument borrows ideas from differential equations, more specifically Gronwall’s inequality. Furthermore, we prove that the expectation value of the photon number operator (\({\mathcal {N}}_\mathrm{ph}\)) on the ground state is bounded and, moreover, that the ground-state vector belongs to the domain of \({\mathcal {N}}^{1 -\epsilon }_\mathrm{ph}\), where \(\epsilon >0\) is as small as we want, depending on the coupling constant.