Spectral Analysis of a Two Body Problem with Zero Range Perturbation

TL;DR

This paper analyzes the spectral properties of a quantum two-body system with zero-range perturbations across different dimensions, providing detailed insights into its point spectrum, asymptotics, and scattering behavior.

Contribution

It offers a comprehensive characterization of the spectrum and scattering properties of a quantum system with zero-range interactions, extending understanding in multiple dimensions.

Findings

Detailed point spectrum characterization

Asymptotic behavior with respect to parameters

Partial description of positive spectrum and scattering

Abstract

We consider a class of singular, zero-range perturbations of the Hamiltonian of a quantum system composed by a test particle and a harmonic oscillators in dimension one, two and three and we study its spectrum. In facts we give a detailed characterization of point spectrum and its asymptotic behavior with respect to the parameters entering the Hamiltonian. We also partially describe the positive spectrum and scattering properties of the Hamiltonian.