Nonexistence of absorbing points for coupling resonance functions

Let $H_0$ and $V$ be the operators specified in the preceding setup, and let an absorbing point mean a point of the resolvent set at which some coupling resonance function tends to infinity along a half-interval ending at that point. Nonexistence conjecture. The pair $H_0$ and $V$ does not have absorbing points, whether isolated or not. This conjecture would rule out the second type of singularity in the preceding classification theorem and further restrict the possible behavior of coupling resonance functions; its resolution is not supplied here.