Perfect lattice perturbation theory: A study of the anharmonic oscillator

As an application of perfect lattice perturbation theory, we construct an O (lambda) perfect lattice action for the anharmonic oscillator analytically in momentum space. In coordinate space, we obtain a set of 2-spin and 4-spi n couplings proportional to lambda, which we evaluate for various masses. T hese couplings never involve variables separated by more than two lattice s pacings. The O(lambda) perfect action is simulated and compared to the stan dard action. We discuss the improvement for the first two energy gaps Delta E-1, Delta E-2 and for the scaling quantity Delta E-2/Delta E-1 in differe nt regimes of the interaction parameter, and of the correlation length. For the quartic oscillator - which corresponds to an asymptotically free theor y - we also discuss a classically perfect action. The single gaps perform v ery well, which corresponds to a clearly improved asymptotic scaling. On th e other hand, it turns out to be difficult to demonstrate an improvement fo r the scaling ratio.