Analyticity of resonances and eigenvalues and spectral properties of the massless spin-boson model (Q1729710)

We mainly reproduce the authors' summary with some additional details. \par The authors analyze the massless Spin-Boson model (a model of quantum field theory). The full Spin-Boson Hamiltonian is defined as $H := H_0 + gV$ for some coupling constant $g \in {\mathbb C}$. The operator $H$ is densely defined on the Hilbert space ${\mathcal H}={\mathbb C}^2\otimes {\mathcal F}\left[ \mathfrak{h}\right]$, where ${\mathcal F}\left[ \mathfrak{h}\right]$ denotes the standard bosonic Fock space, $\mathfrak h:= L^2(\mathbb R^3,{\mathbb C})$. The Hamiltonian $H_0$ is defined as \[ H_0:=K \otimes {\mathbb 1} + {\mathbb 1} \otimes H_f , \qquad K:= \begin{pmatrix} e_1 & 0 \\ 0 & e_0 \end{pmatrix} , \] \[ H_f:=\int \mathrm{d^3}k \, \omega(k) a(k)^* a(k) , \] where $0 = e_0 < e_1$, $\omega(k) = |k|$, and $a, a^*$ are the annihilation and creation operators on standard Fock space. The interaction term reads \[ V:= \sigma_1\otimes \left( a(f) + a(f)^*\right) , \qquad \text{where} \qquad \sigma_1= \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \] and the boson form factor is given by $f: {\mathbb R}^3 \setminus \{0\}\to {\mathbb R}, k\mapsto e^{-\frac{k^2}{\Lambda^2}}|k|^{-\frac{1}{2}+\mu} .$ \par Another object investigated in the paper under review is \textit{complex dilated Hamiltonian} $H^\theta$ that allows to study resonances as eigenvalues. \par ''We extend the method of Pizzo multiscale analysis for resonances introduced by \textit{V. Bach} et al. [Adv. Math. 314, 540--572 (2017; Zbl 1366.81318)] 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 [\textit{A. Pizzo}, Ann. Henri Poincaré 4, No. 3, 439--486 (2003; Zbl 1057.81024)]. 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.''