  This work studies the semiclassical methods in multi-dimensional quantum
systems bounded by finite potentials. By replacing the Maslov index by the
scattering phase, the modified transfer operator method gives rather accurate
corrections to the quantum energies of the circular and square potential pots
of finite heights. The result justifies the proposed scattering phase
correction which paves the way for correcting other semiclassical methods based
on Green functions, like Gutzwiller trace formula, dynamical zeta functions,
and Landauer-B\"uttiker formula.
