Large orders in the 1/N perturbation theory by inverse scattering in one dimension

When one tries to compute large orders in the 1/N series à la Lipatov a complicated non-linear equation for the instanton is found in ø⁴ or non-linear sigma models.We solve here this equation in the one-dimensional case (quantum mechanics) by inverse scattering techniques. From the instanton solutions we obtain theK ^th order of the 1/N perturbation theory up to 0(K ^–1) for the 0(N) symmetric anharmonic oscillator and up to a factor 0(K ⁰) for a non-symmetric model. In the symmetric case we agree with results recently obtained in quantum mechanics by Hikami and Brézin following a different procedure. For the non-symmetric anharmonic oscillator we believe our formulae are new.