  Reflection of charge excitations at the step in the interaction strength in a
Luttinger liquid can be of the Andreev type, even if the interactions are
purely repulsive. The region with stronger repulsion plays the role of a normal
metal in a normal-metal /superconductor junction, whereas the region with
weaker repulsion plays the role of a superconductor. It is shown that this
quasi-Andreev reflection leads to a number of proximity-like effects, including
the local enhancement (suppression) of superconducting fluctuations on the
quasi-normal (quasi-superconducting) side of the step, significant modification
of the local density of states, as well as others. The observable consequences
of these proximity effects are analyzed for the case of single- and
two-particle tunneling from a normal-metal or superconducting tip into an
inhomogeneous Luttinger-liquid wire.
