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.