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 $H_{\alpha }$ of the Laplacian on \(L^2(\mathbb{R}^3)\) and establish Breit--Wigner-type asymptotics for the spectral density of $H_{\alpha }$ along the resonance $\lambda (\alpha )$ near a simple embedded eigenvalue ${\lambda }_0$ of $H_a$ as $\alpha \to a$. We also obtain similar asymptotic behaviour for the scattering cross-section and the average time delay.