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

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 full 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 Robinson (Ann Inst H Poincaré Sect A (N.S.) 21(3):185–215, 1974 ) as well as Radin and Simon (J Differ Equ 29(2):289–296, 1978 ).