Multi-parameter Perturbations of the Laplacian and Resonance Near a Simple Embedded Eigenvalue

Abstract

This paper continues the study of resonance phenomena initiated in [3] for rank-one perturbations. We consider finite-rank multi-parameter perturbations $H_α$ of the Laplacian on \(L^2(\mathbb{R}^3)\) and establish Breit--Wigner-type asymptotics for the spectral density of $H_α$ along the resonance $λ(α)$ near a simple embedded eigenvalue $λ_0$ of $H_a$ as $α\to a$. We also obtain similar asymptotic behaviour for the scattering cross-section and the average time delay.