  We study in this paper the ground state energy of a free bosonic theory on a
finite interval of length $R$ with either a pair of sine-Gordon type or a pair
of Kondo type interactions at each boundary. This problem has potential
applications in condensed matter (current through superconductor-Luttinger
liquid-superconductor junctions) as well as in open string theory (tachyon
condensation). While the application of Bethe ansatz techniques to this problem
is in principle well known, considerable technical difficulties are
encountered. These difficulties arise mainly from the way the bare couplings
are encoded in the reflection matrices, and require complex analytic
continuations, which we carry out in detail in a few cases.
