  The decay rate for a particle in a metastable cubic potential is investigated
in the quantum regime by the Euclidean path integral method in semiclassical
approximation. The imaginary time formalism allows one to monitor the system as
a function of temperature. The family of classical paths, saddle points for the
action, is derived in terms of Jacobian elliptic functions whose periodicity
sets the energy-temperature correspondence. The period of the classical
oscillations varies monotonically with the energy up to the sphaleron, pointing
to a smooth crossover from the quantum to the activated regime. The softening
of the quantum fluctuation spectrum is evaluated analytically by the theory of
the functional determinants and computed at low $T$ up to the crossover. In
particular, the negative eigenvalue, causing an imaginary contribution to the
partition function, is studied in detail by solving the Lam\`{e} equation which
governs the fluctuation spectrum. For a heavvy particle mass, the decay rate
shows a remarkable temperature dependence mainly ascribable to a low lying soft
mode and, approaching the crossover, it increases by a factor five over the
predictions of the zero temperature theory. Just beyond the peak value, the
classical Arrhenius behavior takes over. A similar trend is found studying the
quartic metastable potential but the lifetime of the latter is longer by a
factor ten than in a cubic potential with same parameters. Some formal
analogies with noise-induced transitions in classically activated metastable
systems are discussed.
