Resonance index and singular spectral shift function

This paper is a continuation of my previous work on absolutely continuous and singular spectral shift functions, where it was in particular proved that the singular part of the spectral shift function is an a.e. integer-valued function. It was also shown that the singular spectral shift function is a locally constant function of the coupling constant $r,$ with possible jumps only at resonance points. Main result of this paper asserts that the jump of the singular spectral shift function at a resonance point is equal to the so-called resonance index, --- a new (to the best of my knowledge) notion introduced in this paper. Resonance index can be described as follows. For a fixed $\lambda$ the resonance points $r_0$ of a path $H_r$ of self-adjoint operators are real poles of a certain meromorphic function associated with the triple $(\lambda+i0; H_0,V).$ When $\lambda+i0$ is shifted to $\lambda+iy$ with small $y>0,$ that pole get off the real axis in the coupling constant complex plane and, in general, splits into some $N_+$ poles in the upper half-plane and some $N_-$ poles in the lower half-plane (counting multiplicities). Resonance index of the triple $(\lambda; H_{r_0},V)$ is the difference $N_+-N_-.$ Based on the theorem just described, a non-trivial example of singular spectral shift function is given.