Functional integration through inverse scattering variables. II

Abstract

We continue to develop the method of functional integration over spectral variables introduced in a previous paper. The usual functional integration variables are taken as potentials of an auxiliary linear problem whose spectral data become new integration variables. We deal in this paper with the quantum pendulum and the one- and two-dimensional anharmonic oscillators. We find the functional integration measure and the integration bounds for the spectral variables associated with these quantum systems. This integration measure is valid semiclassically. We compute with it the functional integral for the systems mentioned before. We get in this way integral representations for the ground-state energies. These integral representations turn out to be exact in the semiclassical limit in all cases. They possess the correct large-order behavior of the perturbative expansions (both in Borel-summable and in the non-Borel-summable cases). They also exhibit correctly the tunnel-effect features.