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 the Kth 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(K0) 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. © 1979 Springer-Verlag."