  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].
