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arXiv:1308.2242 · Letters in Mathematical Physics

The Semi-Infinite q-Boson System with Boundary Interaction

Jan Felipe van Diejen, Erdal Emsiz

Published Oct 21, 2013• 9 citations
arXivHTMLPDF
DOI
10.1007/s11005-013-0657-y

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Upon introducing a one-parameter quadratic deformation of the q-boson algebra and a diagonal perturbation at the end point, we arrive at a semi-infinite q-boson system with a two-parameter boundary interaction. The eigenfunctions are shown to be given by Macdonald’s hyperoctahedral Hall–Littlewood functions of type BC. It follows that the n-particle spectrum is bounded and absolutely continuous and that the corresponding scattering matrix factorizes as a product of two-particle bulk and one-particle boundary scattering matrices.
We study the effect of the boundary on a system of weakly interacting bosons in one dimension.It strongly influences the boson density, which is completely suppressed at the boundary position.Away from it, the density is depleted over the distances on the order of the healing length at the mean-field level.Quantum fluctuations modify the density profile considerably.The local density approaches the average one as an inverse square of the distance from the boundary.We calculate an analytic expression for the density profile at arbitrary separations from the boundary.We then consider the problem of localization of a foreign quantum particle (impurity) in the potential created by the inhomogeneous boson density.At the mean-field level, we find exact results for the energy spectrum of the bound states, the corresponding wave functions, and the condition for interaction-induced localization.The quantum contribution to the boson density gives rise to small corrections of the bound-state energy levels.However, it is fundamentally important for the existence of a long-range Casimir-like interaction between the impurity and the boundary.
The Efimov effect (in a broad sense) refers to the onset of a geometric sequence of many-body bound states as a consequence of the breakdown of continuous scale invariance to discrete scale invariance. While originally discovered in three-body problems in three dimensions, the Efimov effect has now been known to appear in a wide spectrum of many-body problems in various dimensions. Here, we introduce a simple, exactly solvable toy model of two identical bosons in one dimension that exhibits the Efimov effect. We consider the situation where the bosons reside on a semi-infinite line and interact with each other through a pairwise δ-function potential with a particular position-dependent coupling strength that makes the system scale invariant. We show that, for sufficiently attractive interaction, the bosons are bound together, and a new energy scale emerges. This energy scale breaks continuous scale invariance to discrete scale invariance and leads to the onset of a geometric sequence of two-body bound states. We also study the two-body scattering off the boundary and derive the exact reflection amplitude that exhibits discrete scale invariance.