  Inhomogeneities and junctions in wires are natural sources of scattering, and
hence resistance. A conducting fixed point usually requires an adiabatically
smooth system. One notable exception is "healing", which has been predicted in
systems with special symmetries, where the system is driven to the homogeneous
fixed point. Here we present theoretical results for a different type of
conducting fixed point which occurs in inhomogeneous wires with an abrupt jump
in hopping and interaction strength. We show that it is always possible to tune
the system to an unstable conducting fixed point which does not correspond to
translational invariance. We analyze the temperature scaling of correlation
functions at and near this fixed point and show that two distinct boundary
exponents appear, which correspond to different effective Luttinger liquid
parameters. Even though the system consists of two separate interacting parts,
the fixed point is described by a single conformally invariant boundary theory.
We present details of the general effective bosonic field theory including the
mode expansion and the finite size spectrum. The results are confirmed by
numerical quantum Monte Carlo simulations on spinless fermions. We predict
characteristic experimental signatures of the local density of states near
junctions.
