本质谱内的谱流V：关于耦合共振吸收点
Spectral flow inside essential spectrum V: on absorbing points of coupling resonances
摘要：
令$H_0$和$V$是自伴算子，使得$V$允许具有有界自伴$J$和$|H_0|^{1/2}$ -紧$F.$耦合共振函数的因子分解$V = F^*JF$，$H_0$和$V$对的$r_j(z),$可以被定义为$r_j(z) = - σ_j(z) ^{-1},$，其中$σ_j(z)$是紧算子值全纯函数$F(H_0-z)^{-1} F^*J.$的特征值。函数$r_j(z)$在~ $H_0.$的预解集上形成一个无穷值全纯函数。这些函数包含关于$H_0, V$对的大量信息（这在秩为1的$V$的情况下是众所周知的）。预解集的一个点$z_0$我们称之为\emph{absorbing}，如果某个$r_j(z)$沿着某个半区间作为$z \to z_0$到达$\infty$。本文给出了关于耦合共振吸收点的部分结果。
英文摘要：
Let $H_0$ and $V$ be self-adjoint operators, such that $V$ admits a factorisation $V = F^*JF$ with bounded self-adjoint $J$ and $|H_0|^{1/2}$-compact $F.$ Coupling resonance functions, $r_j(z),$ of the pair $H_0$ and $V$ can be defined as $r_j(z) = - σ_j(z) ^{-1},$ where $σ_j(z)$ are eigenvalues of the compact-operator valued holomorphic function $F(H_0-z)^{-1} F^*J.$ Taken together, the functions $r_j(z)$ form an infinite-valued holomorphic function on the resolvent set of~$H_0.$ These functions contain a lot of information about the pair $H_0, V$ (this is well-known in the case of rank one $V$). A point $z_0$ of the resolvent set we call \emph{absorbing} if some $r_j(z)$ goes to $\infty$ as $z \to z_0$ along some half-interval. In this note I present some partial results concerning absorbing points of coupling resonances.
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