  We study the ultraviolet asymptotics in $A_n$ affine Toda theories with
integrable boundary actions. The reflection amplitudes of non-affine Toda
theories in the presence of conformal boundary actions have been obtained from
the quantum mechanical reflections of the wave functional in the Weyl chamber
and used for the quantization conditions and ground-state energies. We compare
these results with the thermodynamic Bethe ansatz derived from both the bulk
and (conjectured) boundary scattering amplitudes. The two independent
approaches match very well and provide the non-perturbative checks of the
boundary scattering amplitudes for Neumann and $(+)$ boundary conditions. Our
results also confirm the conjectured boundary vacuum energies and the duality
conjecture between the two boundary conditions.
