We consider splittings of the energies of degenerate quasimodes in a t
wo-dimensional channel. This system is a model for splittings of degen
eracies due to multidimensional tunneling in a general quasi-integrabl
e system. We formulate a determinant quantization condition in terms o
f the transfer matrix, which enables us to relate the splittings to el
ements of the scattering matrix. We find that the splittings can be du
e to either a single- or a multiple-scattering process. In the former
case, the results reduce to an earlier theoretical analysis which used
an assumption that the splittings are canonically invariant quantitie
s. We demonstrate a quantitative agreement between semiclassical theor
y and numerical calculations of the splittings.