  We study the spectrum of the integrable open XXX Heisenberg spin chain
subject to non-diagonal boundary magnetic fields. The spectral problem for this
model can be formulated in terms of functional equations obtained by separation
of variables or, equivalently, from the fusion of transfer matrices. For
generic boundary conditions the eigenvalues cannot be obtained from the
solution of finitely many algebraic Bethe equations. Based on careful finite
size studies of the analytic properties of the underlying hierarchy of transfer
matrices we devise two approaches to analyze the functional equations. First we
introduce a truncation method leading to Bethe type equations determining the
energy spectrum of the spin chain. In a second approach the hierarchy of
functional equations is mapped to an infinite system of non-linear integral
equations of TBA type. The two schemes have complementary ranges of
applicability and facilitate an efficient numerical analysis for a wide range
of boundary parameters. Some data are presented on the finite size corrections
to the energy of the state which evolves into the antiferromagnetic ground
state in the limit of parallel boundary fields.
