Abstract
We investigate the properties of the 1/N-expansion for the quartic anharmonic oscillator in quantum mechanics. The first seven terms of the expansion for the energy ground and first excited levels are obtained analytically. We have found also the large-order behaviour of the 1/N-expansion coefficients in closed form and convinced ourselves that the asymptotic series obtained is Borel summable. We use the formulae derived 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 componentsN. These exact expressions enable us to guess the large-order behaviour of the perturbative coefficients forN=0,1, ..., 4. At the end we give an example of summing the asymptotic series in powers of 1/N applying the Padé-Borel method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ferrell R. A., Scalapino D. J.: Phys. Rev. A9 (1974) 846.
Bray A. J.: J. Phys. A7 (1974) 2144.
Dolgov A. D., Eletzsky V. L., Popov V. S.: Preprint ITEP-72, Moscow, 1979.
Brezin E., Hikami S.: J. Phys. A12 (1979) 759.
Caswell W. E.: Ann. Phys.123 (1979) 153.
Brezin E., LeGuillou J. -C., Zinn-Justin J.: Phys. Rev. D15 (1977) 1544, 1558.
Brezin E., Parisi G., Zinn-Justin J.: Phys. Rev. D16 (1977) 408.
Author information
Authors and Affiliations
Additional information
We thank Drs. D. I. Kazakov, V. K. Mitrjushkin and O. V. Tarasov for their interest and for useful discussions. One of us (M.A.S.) is particularly grateful to Dr. Jiří Niederle for his kind hospitality during the Symposium at Bechyně.
Rights and permissions
About this article
Cite this article
Koudinov, A.V., Smondyrev, M.A. 1/N-expansion for the anharmonic oscillator. Czech J Phys 32, 556–564 (1982). https://doi.org/10.1007/BF01596846
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01596846