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We study the effect of the boundary on a system of weakly interacting bosonsin one dimension. It strongly influences the boson density which is completelysuppressed at the boundary position. Away from it, the density is depleted overthe distances on the order of the healing length at the mean-field level.Quantum fluctuations modify the density profile considerably. The local densityapproaches the average one as an inverse square of the distance from theboundary. We calculate an analytic expression for the density profile atarbitrary separations from the boundary. We then consider the problem oflocalization of a foreign quantum particle (impurity) in the potential createdby the inhomogeneous boson density. At the mean-field level, we find exactresults for the energy spectrum of the bound states, the corresponding wavefunctions, and the condition for interaction-induced localization. The quantumcontribution to the boson density gives rise to small corrections of the boundstate energy levels. However, it is fundamentally important for the existenceof a long-range Casimir-like interaction between the impurity and the boundary.

Density profile of a semi-infinite one-dimensional Bose gas and bound states of the impurity