论文标题
$ 2 \ times 2 $运算符矩阵的频谱分析。离散频谱渐近学
Analysis of the spectrum of a $2\times 2$ operator matrix. Discrete spectrum asymptotics
论文作者
论文摘要
我们考虑$ 2 \ times 2 $运算符矩阵$ {\ Mathcal a}_μ，$ $ $ $μ> 0 $与晶格系统相关，这些晶格系统描述了两个相同的玻色子和一个粒子，一种互动中的另一个粒子，而没有粒子数量的保护。我们获得了FADDEEV方程的类似物及其对称版本的$ {\ Mathcal a} _ $ $ $的对称版本。我们通过广义的Friedrichs模型的频谱来描述$ {\ Mathcal a}_μ$的基本频谱的新分支。可以确定的是，$ {\ Mathcal a}_μ$的基本频谱最多是三个有界闭合间隔的结合，并研究了它们的位置。对于临界值$μ_0$的耦合常数$μ$，我们建立了无限的许多特征值的存在，这些特征值位于$ {\ Mathcal a}_μ$的基本频谱的两侧。在这种情况下，找到了$ {\ mathcal a}_μ$离散频谱的渐近公式。
We consider a $2 \times 2$ operator matrix ${\mathcal A}_μ,$ $μ>0$ related with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We obtain an analogue of the Faddeev equation and its symmetric version for the eigenfunctions of ${\mathcal A}_μ$. We describe the new branches of the essential spectrum of ${\mathcal A}_μ$ via the spectrum of a family of generalized Friedrichs models. It is established that the essential spectrum of ${\mathcal A}_μ$ consists the union of at most three bounded closed intervals and their location is studied. For the critical value $μ_0$ of the coupling constant $μ$ we establish the existence of infinitely many eigenvalues, which are located in the both sides of the essential spectrum of ${\mathcal A}_μ$. In this case, an asymptotic formula for the discrete spectrum of ${\mathcal A}_μ$ is found.
