  A newly developed method for systematically improving the convergence of path
integrals for transition amplitudes, introduced in Phys. Rev. Lett. 94 (2005)
180403, Phys. Rev. B 72 (2005) 064302, Phys. Lett. A 344 (2005) 84, and
expectation values, introduced in Phys. Lett. A 360 (2006) 217, is here applied
to the efficient calculation of energy spectra. We show how the derived
hierarchies of effective actions lead to substantial speedup of the standard
path integral Monte Carlo evaluation of energy levels. The general results and
the ensuing increase in efficiency of several orders of magnitude are shown
using explicit Monte Carlo simulations of several distinct models.
