We use first-order perturbation theory near the fermionic limit of the δ-function Bose gas in one dimension (i.e. a system of weakly interacting fermions) to study three situations of physical interest. The calculation is done using a pseudopotential which takes the form of a two-body δ''-function interaction. The three cases considered are the behaviour of the system with a hard wall, with a point where the strength of the pseudopotential changes discontinuously, and with a region of finite length where the pseudopotential strength is non-zero (this is sometimes used as a model for a quantum wire). In all cases, we obtain exact expressions for the density to first order in the pseudopotential strength. The asymptotic behaviour of the densities is in agreement with the results obtained from bosonization for a Tomonaga–Luttinger liquid, namely, an interaction dependent power-law decay of the density far from the hard wall, a reflection from the point of discontinuity and transmission resonances for the interacting region of finite length. Our results provide a non-trivial verification of the Tomonaga–Luttinger liquid description of the δ-function Bose gas near the fermionic limit.

71.10.Pm Fermions in reduced dimensions (anyons, composite fermions, Luttinger liquid, etc.)

05.30.Jp Boson systems

05.30.Fk Fermion systems and electron gas

82B10 Quantum equilibrium statistical mechanics (general)

Quantum gases, liquids and solids

Condensed matter: electrical, magnetic and optical

Statistical physics and nonlinear systems

Issue 27 (11 July 2003)

Received 15 January 2003 , in final form 21 May 2003

Published 25 June 2003