Spectral flow inside essential spectrum V: on absorbing points of coupling resonances

Let $H_0$ and $V$ be self-adjoint operators, such that $V$ admits a factorisation $V=F^{\ast }JF$ with bounded self-adjoint $J$ and $\mathrm{\mid }{H}_0{\mathrm{\mid }}^{1\mathrm{/}2}$-compact $F\mathrm{.}$ Coupling resonance functions, $r_j(z),$ of the pair $H_0$ and $V$ can be defined as $r_j(z)=-{\sigma }_j(z{)}^{-1},$ where ${\sigma }_j(z)$ are eigenvalues of the compact-operator valued holomorphic function $F(H_0-z{)}^{-1}{F}^{\ast }J\mathrm{.}$ Taken together, the functions $r_j(z)$ form an infinite-valued holomorphic function on the resolvent set of~$H_0\mathrm{.}$ 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 $\mathrm{\infty }$ as $z\to z_0$ along some half-interval. In this note I present some partial results concerning absorbing points of coupling resonances.