On Lennard-Jones-type potentials on the half-line

TL;DR

This paper investigates the spectral and scattering properties of a quantum particle under Lennard-Jones-type potentials on the half-line, addressing the mathematical challenges posed by the potential's singularity at the origin.

Contribution

It provides a rigorous construction of the Hamiltonian for Lennard-Jones potentials on the half-line and extends previous results on spectral and scattering analysis.

Findings

Constructed the Hamiltonian via quadratic forms for singular potentials

Analyzed spectral properties of the Lennard-Jones potential

Extended classical results to include these potentials

Abstract

In this paper we study a particle under the influence of a Lennard-Jones potential moving in a simple quantum wire modelled by the positive half-line. Despite its physical significance, this potential is only rarely studied in the literature and due to its singularity at the origin it cannot be considered as a standard perturbation of the one-dimensional Laplacian. It is therefore our aim to provide a thorough description of the Hamiltonian in one dimension via the construction of a suitable quadratic form. Our results include a discussion of spectral and scattering properties which finally allows us to generalise some results from [Robinson1974] as well as [RadinSimon1978].