Valley approximation for the Borel function

A simple one-dimensional integral is investigated as a model for large-order estimation of the perturbative expansion in quantum mechanics with degenerate vacua. A Borel function analysis allows us to separate nonperturbative contributions from perturbative ones. Issues such as the cancellation between the perturbative and nonperturbative contributions of ambiguity due to non-Borel summability and the large-order estimation in terms of a dispersion integral are discussed. A stationary-point approximation for the Borel function is proposed to connect the simple integral to the quantum-mechanical case based on the new valley trajectory, which was recently formulated