Abstract

We study an unbounded \(2{\times}2\) operator matrix \( \mathcal{A} \) in the direct product of two Hilbert spaces. We obtain asymptotic formulas for the number of eigenvalues of \( \mathcal{A} \). We consider a \(2{\times}2\) operator matrix \( \mathcal{A} _\mu\), where \(\mu>0\) is the coupling constant, associated with the Hamiltonian of a system with at most three particles on the lattice \( \mathbb{Z} ^3\). We find the critical value \(\mu_0\) of the coupling constant \(\mu\) for which \( \mathcal{A} _{\mu_0}\) has an infinite number of eigenvalues. These eigenvalues accumulate at the lower and upper bounds of the essential spectrum. We obtain an asymptotic formula for the number of such eigenvalues in both the left and right parts of the essential spectrum.

中文翻译：

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$$ 2 {\ times} 2 $$算子矩阵的无穷多个特征值：渐近离散谱

摘要

我们在两个希尔伯特空间的直接积中研究无界\（2 {\ times} 2 \）算子矩阵\（\ mathcal {A} \）。我们获得\（\ mathcal {A} \）的特征值数量的渐近公式。我们考虑一个\（2 {\ times} 2 \）算子矩阵\（\ mathcal {A} _ \ mu \），其中\（\ mu> 0 \）是耦合常数，与具有晶格\（\ mathbb {Z} ^ 3 \）上最多三个粒子。我们找到耦合常数\（\ mu \）的临界值\（\ mu_0 \），其中\（\ mathcal {A} _ {\ mu_0} \）具有无限数量的特征值。这些特征值累积在基本频谱的下限和上限。我们获得了在基本频谱的左侧和右侧部分中此类特征值数量的渐近公式。