Ground State and Resonances in the Standard Model of the Non-Relativistic QED

Deckert

Hänle

2019

Commun. Math. Phys.

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 .

Section: Introductionmentioning

confidence: 99%

Relation Between the Resonance and the Scattering Matrix in the Massless Spin-Boson Model

Deckert

Hänle

2019

Commun. Math. Phys.

“…In the form needed in the analysis of quantum systems with infinitely many degrees of freedom, these tools were first introduced in [10] and systematically developed in [8], [11] and [6]. Important refinements of these methods have appeared in [29], [30], [33], [43], [3], [22]; and references given there. Some alternative methods have been introduced in [1] and [4].…”

Section: Introductionmentioning

confidence: 99%

Quantum Electrodynamics of Atomic Resonances

Faupin

Fröhlich

et al. 2015

Commun. Math. Phys.

Abstract. A simple model of an atom interacting with the quantized electromagnetic field is studied. The atom has a finite mass m, finitely many excited states and an electric dipole moment, d0 = −λ0 d, where d i = 1, i = 1, 2, 3, and λ0 is proportional to the elementary electric charge. The interaction of the atom with the radiation field is described with the help of the Ritz Hamiltonian, − d0 · E, where E is the electric field, cut off at large frequencies. A mathematical study of the Lamb shift, the decay channels and the life times of the excited states of the atom is presented. It is rigorously proven that these quantities are analytic functions of the momentum p of the atom and of the coupling constant λ0, provided | p| < mc and |ℑ p| and |λ0| are sufficiently small. The proof relies on a somewhat novel inductive construction involving a sequence of 'smooth Feshbach-Schur maps' applied to a complex dilatation of the original Hamiltonian, which yields an algorithm for the calculation of resonance energies that converges super-exponentially fast.

“…[21,22,5,4]) and the renormalization group method (see e.g. [7,9,8,6,10,2,15,16,25,14,12]). In both cases, a family of spectrally dilated Hamiltonians is analyzed since this allows for complex eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, resolvent and spectral estimates were obtained therein. Based on this new method, several simplifications and applications were developed in a variety of works [6,10,12,16,15,2,12,25,14,2]. The Pizzo multiscale analysis was first invented in [21,22] and then adapted in order to gain access to spectral and resolvent estimates and the construction of ground-states in [4,5].…”

Section: Introductionmentioning

confidence: 99%

Analyticity of resonances and eigenvalues and spectral properties of the massless Spin–Boson model

Journal of Functional Analysis

Deckert

Hänle

2019

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