Coupled Anharmonic Oscillators. I. Equal-Mass Case

This is the first in a series of papers on the large-order behavior of perturbation theory for coupled anharmonic oscillators. We exploit previously published dispersion techniques to convert the calculation of perturbation theory in large order into a barrier-penetration problem. We then introduce new semiclassical methods for describing tunneling through nonspherically symmetric, N-dimensional potentials. To illustrate our new methods, we calculate the large-order behavior of perturbation theory for a simple system of two equal-mass oscillators with quartic coupling. Our predictions are in complete agreement with computer calculations. We then extend our results to oscillators with x^2N coupling, N-oscillator systems, and some infinite-oscillator systems.