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

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.