论文标题
$ 2 \ times 2 $运算符矩阵的离散特征值
Discrete Eigenvalues of a $2 \times 2$ Operator Matrix
论文作者
论文摘要
我们认为$ 2 \ times2 $ block运算符矩阵$ {\ mathcal a}_μ$ $ $ $ $（$$μ> 0 $是耦合常数$）$以直接的总和为Bosonic Fock空间的单粒子子空间。描述了$ {\ mathcal a} _ $的基本频谱的位置，并估计其边界。结果表明，存在关键值$μ_l^0（γ）$，带有$γ> 0 $和$μ_r^0（γ）$，$γ<12 $的耦合常数$μ> 0 $，以便所有$γ> 0 $ $ $ $ $（γ<12）$ $（μ=μ_r^0（γ）$在其基本频谱的l.h.s. $（$ r.h.s。$）上具有无限的特征值。我们证明了所有$μ\ not \ in \ in \ in \ {μ_l^0（γ），γ），μ_r^0（μ_r^0（γ）$ a的$ a n oterator oterator a plastorator otatortor L.H.S.和R.H.S.的许多离散值有限。
We consider a $2\times2$ block operator matrix ${\mathcal A}_μ$ $($$μ>0$ is a coupling constant$)$ acting in the direct sum of one- and two-particle subspaces of a bosonic Fock space. The location of the essential spectrum of ${\mathcal A}_μ$ is described and its bounds are estimated. It is shown that there exist the critical values $μ_l^0(γ)$ with $γ>0$ and $μ_r^0(γ)$ with $γ<12$ of the coupling constant $μ>0$ such that for all $γ>0$ $(γ<12)$ the operator ${\mathcal A}_μ$ with $μ=μ_l^0(γ)$ $(μ=μ_r^0(γ)$ has infinitely many eigenvalues on the l.h.s. $($r.h.s.$)$ of the its essential spectrum. We prove that for all $μ\not\in \{μ_l^0(γ),μ_r^0(γ)\}$ the operator ${\mathcal A}_μ$ has finitely many discrete eigenvalues on the l.h.s. and r.h.s. of its essential spectrum.
