We extend the method of Pizzo multiscale analysis for resonances introduced in [5] in order to infer analytic properties of resonances and eigenvalues (and their eigenprojections) as well as estimates for the localization of the spectrum of dilated Hamiltonians and norm-bounds for the corresponding resolvent operators, in neighborhoods of resonances and eigenvalues. We apply our method to the massless Spin-Boson model assuming a slight infrared regularization. We prove that the resonance and the ground-state eigenvalue (and their eigenprojections) are analytic with respect to the dilation parameter and the coupling constant. Moreover, we prove that the spectrum of the dilated Spin-Boson Hamiltonian in the neighborhood of the resonance and the ground-state eigenvalue is localized in two cones in the complex plane with vertices at the location of the resonance and the ground-state eigenvalue, respectively. Additionally, we provide norm-estimates for the resolvent of the dilated Spin-Boson Hamiltonian near the resonance and the ground-state eigenvalue. The topic of analyticity of eigenvalues and resonances has let to several studies and advances in the past. However, to the best of our knowledge, this is the first time that it is addressed from the perspective of Pizzo multiscale analysis. Once the multiscale analysis is set up our method gives easy access to analyticity: Essentially, it amounts to proving it for isolated eigenvalues only and use that uniform limits of analytic functions are analytic. The type of spectral and resolvent estimates that we prove are needed to control the time evolution including the scattering regime. The latter will be demonstrated in a forthcoming publication. The introduced multiscale method to study spectral and resolvent estimates follows its own inductive scheme and is independent (and different) from the method we apply to construct resonances. * miguel.ballesteros@iimas.unam.mx, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autánoma de México

We establish the precise relation between the integral kernel of the scattering matrix and the resonance in the massless Spin-Boson model which describes the interaction of a two-level quantum system with a second-quantized scalar field. For this purpose, we derive an explicit formula for the two-body scattering matrix. We impose an ultraviolet cut-off and assume a slightly less singular behavior of the boson form factor of the relativistic scalar field but no infrared cut-off. The purpose of this work is to bring together scattering and resonance theory and arrive at a similar result as provided by Simon in [39], where it was shown that the singularities of the meromorphic continuation of the integral kernel of the scattering matrix are located precisely at the resonance energies. The corresponding problem has been open in quantum field theory ever since. To the best of our knowledge, the presented formula provides the first rigorous connection between resonance and scattering theory in the sense of [39] in a model of quantum field theory. * miguel.ballesteros@iimas.unam.mx, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autánoma de México θ ∈ D(0, π/16) the operators H θ are densely defined and closed. Moreover, the analytic properties of this family of operators in g and θ are known: Lemma 1.4. The family H θ θ∈R of unitary equivalent, self-adjoint operators with D(H θ ) = D(H) extends to an analytic family of type A for θ ∈ D(0, π/16).The above result was proven for the Pauli-Fierz model in [4, Theorem 4.4], and with small effort that proof can be adapted to our setting. Lemma 1.5. Let θ ∈ C. Then, σ(H θ 0 ) = e i + e −θ r : r ≥ 0, i = 0, 1 .

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In scattering experiments, physicists observe so-called resonances as peaks at certain energy values in the measured scattering cross sections per solid angle. These peaks are usually associate with certain scattering processes, e.g., emission, absorption, or excitation of certain particles and systems. On the other hand, mathematicians define resonances as poles of an analytic continuation of the resolvent operator through complex dilations. A major challenge is to relate these scattering and resonance theoretical notions, e.g., to prove that the poles of the resolvent operator induce the above mentioned peaks in the scattering matrix. In the case of quantum mechanics, this problem was addressed in numerous works that culminated in Simon's seminal paper [33] in which a general solution was presented for a large class of pair potentials. However, in quantum field theory the analogous problem has been open for several decades despite the fact that scattering and resonance theories have been well-developed for many models. In certain regimes these models describe very fundamental phenomena, such as emission and absorption of photons by atoms, from which quantum mechanics originated. In this work we present a first non-perturbative formula that relates the scattering matrix to the resolvent operator in the massless Spin-Boson model. This result can be seen as a major progress compared to our previous works [13] and [12] in which we only managed to derive a perturbative formula. processes, and thus, prove an analogous result that was obtained by Simon in [33] for the N-body Schrödinger operator in this particular model of quantum field theory. More precisely, we show that the pole of a meromorphic continuation of the integral kernel of the scattering matrix is located precisely at the resonance energy. The objective in this result is to contribute to the understanding of the relation between resonance and scattering theory. In our previous works [13] and [12], we were already able to derive perturbative results of this kind in case of the massless and massive Spin-Boson models, respectively. However, both results are only given in leading order with respect to the coupling constant. The present work can be seen as a major improvement of these pertubative results because it provides a closed and non-perturbative formula that connects the integral kernel of the scattering matrix elements for one-boson processes in terms of the dilated resolvent.Our results are based on the well-established fields of scattering and resonance theories and the numerous works in the classical literature of which we want to give a short overview here. Resonance theory, in the realm of quantum field theory, has been developed in a variety of models; see, e.g., [6,8,7,4,9,1,26,27,32,21,15,28,29,2,3,14]. In these works, several techniques have been invented for massless models of quantum field theory in order to cope with the absence of a spectral gap. Scattering theory has also been developed in various models of quantum field theory (see, e....