Pseudoparticle contributions to the energy spectrum of a one-dimensional system

We show that classical solutions of the Euclidean action can be used to calculate the shift in energy levels due to tunneling through a potential barrier. In particular, we use the path integral to compute the kernel of the double-well anharmonic oscillator for a large, but finite, Euclidean time interval by expanding about pseudoparticle solutions (i.e., the kink). This allows us to determine the ground-state energy plus that of the first excited state (the splitting is due to barrier penetration). We find that not only the classical solution must be expanded about, but also nearly stationary trajectories corresponding to kink plus kink-antikink pairs. The quasitranslational invariance must also be dealt with carefully. We compare with the WKB result and find our result more accurate, because it avoids the errors introduced by the linear (Airy functions) connecting formulas.