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Given a self-adjoint operator $H_0$ and a relatively $H_0$-compactself-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 alot of important information about the pair $H_0$ and $V.$ We call themcoupling resonances. In case of rank one (and positive) perturbation $V,$ thereis only one coupling resonance function, which is a Herglotz function. Thiscase has been studied in depth in the literature, and appears in differentsituations, such as Sturm-Liouville theory, random Schr\"odinger operators,harnomic and spectral analyses, etc. The general case is complicated by thefact that the resonance functions are no longer single valued holomorphicfunctions, and potentially can have quite an erratic behaviour, typical forinfinitely-valued holomorphic functions. Of special interest are those couplingresonance functions $r_z$ which approach a real number $r_{\lambda+i0}$ fromthe interval $[0,1]$ as the spectral parameter $z=\lambda+iy$ approaches apoint $\lambda$ of the essential spectrum, since they are responsible forspectral flow through $\lambda$ inside essential spectrum when $H_0$ getsdeformed 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 limitingabsorption principle, then the coupling resonance functions are well-behavednear the essential spectrum in the following sense. Let $I$ be an open intervalinside the essential spectrum of $H_0$ and $\epsilon>0.$ Then there exists acompact subset~$K$ of~$I$ such that $| I \setminus K | < \epsilon,$ and $K$ hasa "non-tangential" neighbourhood in the upper complex half-plane, such that anycoupling resonance function is either single-valued in the neighbourhood, ordoes not take a real value in the interval $[0,1].$

Spectral flow inside essential spectrum III: coupling resonances near essential spectrum