  In this second part of the treatment of instantons in quantum mechanics, the
focus is on specific calculations related to a number of quantum mechanical
potentials with degenerate minima. We calculate the leading multi-instanton
constributions to the partition function, using the formalism introduced in the
first part of the treatise [J. Zinn-Justin and U. D. Jentschura, e-print
quant-ph/0501136]. The following potentials are considered: (i) asymmetric
potentials with degenerate minima, (ii) the periodic cosine potential, (iii)
anharmonic oscillators with radial symmetry, and (iv) a specific potential
which bears an analogy with the Fokker-Planck equation. The latter potential
has the peculiar property that the perturbation series for the ground-state
energy vanishes to all orders and is thus formally convergent (the ground-state
energy, however, is nonzero and positive). For the potentials (ii), (iii), and
(iv), we calculate the perturbative B-function as well as the instanton
A-function to fourth order in g. We also consider the double-well potential in
detail, and present some higher-order analytic as well as numerical
calculations to verify explicitly the related conjectures up to the order of
three instantons. Strategies analogous to those outlined here could result in
new conjectures for problems where our present understanding is more limited.
