Scattering of particles bounded to an infinite planar curve

TL;DR

This paper analyzes quantum particles confined to an infinite planar curve with attractive contact interactions, demonstrating spectral properties and wave operator existence relevant to nanostructure models.

Contribution

It establishes the absolutely continuous spectrum for particles on such curves and proves the existence of wave operators for the associated Hamiltonians.

Findings

Interval between bound state energy and zero is in the spectrum

Wave operators exist for the specified energy interval

Spectrum includes possible embedded eigenvalues

Abstract

Non-relativistic quantum particles bounded to a curve in R^2 by attractive contact $\delta$-interaction are considered. The interval between the energy of the transversal bound state and zero is shown to belong to the absolutely continuous spectrum, with possible embedded eigenvalues. The existence of the wave operators is proved for the mentioned energy interval using the Hamiltonians with the interaction supported by the straight lines as the free ones. Their completeness is not proved. The curve is assumed C^3-smooth, non-intersecting, unbounded, asymptotically approaching two different half-lines (non-parallel or parallel but excluding the "U-case"). Physically, the system can be considered as a model of long nanostructural channel.