Perturbation theory series in quantum mechanics: Phase transition and exact asymptotic forms for the expansion coefficients

We consider the model of a harmonic oscillator with a power-law potential and derive new asymptotic formulas for the coefficients of the perturbation theory series in powers of the coupling constant in the case of a power-law perturbing potential |x|^p, p > 0. We prove the existence of a critical value p ₀ and compute it. It is a threshold in the sense that the asymptotic forms of the studied coefficients for 0 < p < p ₀ and for p > p ₀ differ qualitatively. We note that the considered physical system undergoes a phase transition at p = p ₀ . The analysis uses the Laplace method for functional integrals with Gaussian measures.