1/N-expansion for the anharmonic oscillator

The properties of the 1/N-expansion for the anharmonic oscillator in quantum mechanics have been investigated. The first seven terms of the expansion for the energy of ground and first excited levels are obtained analytically. The high-order behaviour of the 1/N-expansion coefficients in closed form was found, the asymptotic series obtained being Borel summable. The formulae derived was used to find the first seven coefficients of the perturbative expansion in powers of the coupling constant in the case of the double-well potential for arbitrary number of components N. These exact expressions enable us to guess the large-order behaviour of the perturbative coefficients for N=0, 1, ... 4. An example of summing the asymptotic series in powers of 1/N applying the Pade-Borel method is given