Coupling resonances and spectral properties of the product of resolvent and perturbation

Abstract

Given a self-adjoint operator H0 and a relatively compact self-adjoint perturbation V, we study in some detail the spectral properties of the product (H0−z)−1V, z∈C. The eigenvalues of this product are related simply to those of the corresponding product in the situation when the operator H0 is perturbed to H0+sV for s∈C. For some numbers rz, the eigenvalues of (H0+sV−z)−1V, s∈C, are (s−rz)−1. We study the root spaces of the eigenvalues (s−rz)−1 and complex-analytic properties of the functions rz such as branching points. In particular, for a generic case, we give a variety of necessary and sufficient conditions for branching. The functions rz, called coupling resonances, are important in the spectral analysis of H0+rV for any real number r∈R. For instance, they afford a description of the spectral shift function (SSF) of the pair H0 and V, as well as the absolutely continuous and singular parts of the SSF. A thorough study of real-valued coupling resonances rλ for real λ outside of the essential spectrum was carried out in a recent work by the first author. Here we extend this study to the complex domain, motivated by the fact, which is well known in the case of a rank-one perturbation, that the behaviour of coupling resonances rz near the essential spectrum provides valuable information about the latter.