Abstract

Let H0 be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and V be a compact perturbation. We relate the regularity properties of V to various spectral properties of the perturbed operator H0+V. The structures of the discrete spectrum and the embedded eigenvalues are analyzed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results scattered throughout the literature are recovered and extended. For illustrative purposes, the case of the one-dimensional discrete Laplacian is emphasized.