  Given a self-adjoint operator $H_0$ and a relatively $H_0$-compact
self-adjoint operator $V,$ the functions $r_j(z) = - \sigma_j^{-1}(z),$ where
$\sigma_j(z)$ are eigenvalues of the compact operator $(H_0-z)^{-1}V,$ bear a
lot of important information about the pair $H_0$ and $V.$ We call them
coupling resonances. In case of rank one (and positive) perturbation $V,$ there
is only one coupling resonance function, which is a Herglotz function. This
case has been studied in depth in the literature, and appears in different
situations, such as Sturm-Liouville theory, random Schr\"odinger operators,
harnomic and spectral analyses, etc. The general case is complicated by the
fact that the resonance functions are no longer single valued holomorphic
functions, and potentially can have quite an erratic behaviour, typical for
infinitely-valued holomorphic functions. Of special interest are those coupling
resonance functions $r_z$ which approach a real number $r_{\lambda+i0}$ from
the interval $[0,1]$ as the spectral parameter $z=\lambda+iy$ approaches a
point $\lambda$ of the essential spectrum, since they are responsible for
spectral flow through $\lambda$ inside essential spectrum when $H_0$ gets
deformed to $H_1 = H_0+V$ via the path $H_0 + rV, r \in [0,1].$
  In this paper it is shown that if the pair $H_0,$ $V$ satisfies the limiting
absorption principle, then the coupling resonance functions are well-behaved
near the essential spectrum in the following sense. Let $I$ be an open interval
inside the essential spectrum of $H_0$ and $\epsilon>0.$ Then there exists a
compact subset~$K$ of~$I$ such that $| I \setminus K | < \epsilon,$ and $K$ has
a "non-tangential" neighbourhood in the upper complex half-plane, such that any
coupling resonance function is either single-valued in the neighbourhood, or
does not take a real value in the interval $[0,1].$
