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).
KeywordsOperator theory Self-adjoint realisation Quadratic form
Mathematics Subject Classification47A10 47A40 47A15 81V99
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The authors are very happy to thank R. Weder (Universidad Nacional Autónoma de México) for many helpful comments on the manuscript. JK also wants to thank S. Egger (Technion, Israel) for helpful discussions. We also want to thank the referee for pointing out interesting references.