The Spin-Boson model describes a two-level quantum system that interacts with a second-quantized boson scalar field. Recently the relation between the integral kernel of the scattering matrix and the resonance in this model has been established in [14] for the case of massless bosons. In the present work, we treat the massive case. On the one hand, one might rightfully expect that the massive case is easier to handle since, in contrast to the massless case, the corresponding Hamiltonian features a spectral gap. On the other hand, it turns out that the non-zero boson mass introduces a new complication as the spectrum of the complex dilated, free Hamiltonian exhibits lines of spectrum attached to every multiple of the boson rest mass energy starting from the ground and excited state energies. This leads to an absence of decay of the corresponding complex dilated resolvent close to the real line, which, in [14], was a crucial ingredient to control the time evolution in the scattering regime. With the new strategy presented here, we provide a proof of an analogous formula for the scattering kernel as compared to the massless case and use the opportunity to provide the required spectral information by a Mourre theory argument combined with a suitable application of the Feshbach-Schur map instead of complex dilation.

中文翻译：

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大质量自旋玻色子模型中的单玻色子散射过程

自旋玻色子模型描述了一个与二次量子化玻色子标量场相互作用的双能级量子系统。最近在[14]中已经建立了散射矩阵的积分核与该模型中的共振之间的关系，用于无质量玻色子的情况。在目前的工作中，我们处理大规模案例。一方面，人们可能理所当然地期望有质量的情况更容易处理，因为与无质量的情况相比，相应的哈密顿量具有光谱间隙。另一方面，事实证明，非零玻色子质量引入了一个新的复杂性，因为复杂的膨胀的自由哈密顿量的光谱表现出附加到从基态和激发态开始的玻色子静止质量能量的每个倍数的谱线能量。这导致接近实线的相应复杂膨胀溶剂没有衰减，在 [14] 中，这是控制散射区时间演变的关键因素。使用此处介绍的新策略，我们提供了散射核与无质量情况相比的类似公式的证明，并利用机会通过 Mourre 理论论证结合 Feshbach-Schur 的适当应用提供所需的光谱信息映射而不是复杂的扩张。