论文标题
$ 3 \ times 3 $操作员矩阵的离散频谱的分析
Analysis of the discrete spectrum of the family of $3 \times 3$ operator matrices
论文作者
论文摘要
We consider the family of $3 \times 3$ operator matrices ${\bf H}(K),$ $K \in {\Bbb T}^3:=(-π; π]^3$ associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set $Λ\subset {\Bbb t}^3 $证明存在$ {\ bf h}（k）$的许多特征价值，当时的所有$ k \ inλ$当相关的弗里德里希斯模型具有零能量共鸣时，发现每个$ k \ inλ，$ k）$ n数字$ n（k，z）$ bf of e eigenval of eigenval f of eigenval f of eigenval of of eigenval { $ z，$ $ z <0，$满足渐近关系$ \ lim \ limits_ {z \ to -0} n（k，k，z）| \ log | z | z || ||^{ - 1} = {\ Mathcal u} _0 $ with $ 0 <{\ Mathcal U}对于任何$ k \inλ$，如果相关的Friedrichs模型的特征值为零，则操作员$ {\ bf h}（k）$具有有限数量的负特征值，或者是零是正常类型的点点，用于正定义确定的弗里德里奇模型。
We consider the family of $3 \times 3$ operator matrices ${\bf H}(K),$ $K \in {\Bbb T}^3:=(-π; π]^3$ associated with the lattice systems describing two identical bosons and one particle, another nature in interactions, without conservation of the number of particles. We find a finite set $Λ\subset {\Bbb T}^3$ to prove the existence of infinitely many eigenvalues of ${\bf H}(K)$ for all $K \in Λ$ when the associated Friedrichs model has a zero energy resonance. It is found that for every $K \in Λ,$ the number $N(K, z)$ of eigenvalues of ${\bf H}(K)$ lying on the left of $z,$ $z<0,$ satisfies the asymptotic relation $\lim\limits_{z \to -0} N(K, z) |\log|z||^{-1}={\mathcal U}_0$ with $0<{\mathcal U}_0<\infty,$ independently on the cardinality of $Λ.$ Moreover, we prove that for any $K \in Λ$ the operator ${\bf H}(K)$ has a finite number of negative eigenvalues if the associated Friedrichs model has a zero eigenvalue or a zero is the regular type point for positive definite Friedrichs model.
