2021/10/30 by Nurulla Azamov, Azamov, Nurulla, Tom Daniels +1
Mathematics · Physics and Astronomy · #81U24 #FOS: Mathematics #Primary 47A11 #Quantum chaos and dynamical systems #Random Matrices and Applications #Secondary 47A55 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2111.00225
openalex publication_date 2021/10/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. For some numbers rz, the eigenvalues of (H0 + s V - z)-1V are (s - rz)-1, where s is any complex number. 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 + r V for any real number 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.