We study the spectrum of the spin-boson Hamiltonian with two bosons for arbitrary coupling [Formula: see text] in the case when the dispersion relation is a bounded function. We derive an explicit description of the essential spectrum which consists of the so-called two- and three-particle branches that can be separated by a gap if the coupling is sufficiently large. It turns out, that depending on the location of the coupling constant and the energy level of the atom (w.r.t. certain constants depending on the maximal and the minimal values of the boson energy) as well as the validity or the violation of the infrared regularity type conditions, the essential spectrum is either a single finite interval or a disjoint union of at most six finite intervals. The corresponding critical values of the coupling constant are determined explicitly and the asymptotic lengths of the possible gaps are given when [Formula: see text] approaches to the respective critical value. Under minimal smoothness and regularity conditions on the boson dispersion relation and the coupling function, we show that discrete eigenvalues can never accumulate at the edges of the two-particle branch. Moreover, we show the absence of the discrete eigenvalue accumulation at the edges of the three-particle branch in the infrared regular case.

中文翻译：

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任意耦合和有界色散关系的具有两个玻色子的自旋玻色子哈密顿量的光谱分析

在色散关系是有界函数的情况下，我们研究了具有两个玻色子的自旋玻色子哈密顿量的光谱，用于任意耦合[公式：见正文]。我们得出了对基本光谱的明确描述，该光谱由所谓的两粒子和三粒子分支组成，如果耦合足够大，这些分支可以被间隙分开。事实证明，这取决于耦合常数的位置和原子的能级（某些常数取决于玻色子能量的最大值和最小值）以及红外规律类型的有效性或违反条件下，基本谱要么是单个有限区间，要么是最多六个有限区间的不相交并集。耦合常数的相应临界值被明确确定，并且当[公式：见文本]接近各自的临界值时，给出可能间隙的渐近长度。在玻色子色散关系和耦合函数的最小平滑度和规则性条件下，我们表明离散特征值永远不会在双粒子分支的边缘累积。此外，我们表明在红外常规情况下，三粒子分支边缘不存在离散特征值累积。我们证明离散特征值永远不会在双粒子分支的边缘累积。此外，我们表明在红外常规情况下，三粒子分支边缘不存在离散特征值累积。我们证明离散特征值永远不会在双粒子分支的边缘累积。此外，我们表明在红外常规情况下，三粒子分支边缘不存在离散特征值累积。