Quantum-mechanical oscillator with arbitrary anharmonicity: $1/N$ Expansion and perturbation theory

The properties of the $1/N$ expansion are investigated for the problem of an Ndimensional anharmonic oscillator with arbitrary power anharmonieity. The first six terms in the expansion of the energies of the ground and first excited levels are obtained in analytic form. The asymptotic behavior of the coefficients in large orders of the $1/N$ expansion is investigated. The obtained formulas are used to determine expressions for the first six coefficients of the standard perturbation theory in powers of the coupling constant in the case of an $N$-dimensional potential with two degenerate minima. The asymptotic behavior of these coefficients at high orders of perturbation theory is discussed.