Abstract
The inverse scattering transformation (IST) with the angular momentum (λ) as a spectral variable turned out to be a useful method to deal with rotationally invariant problems in field theory at higher spatial dimensions [Refs. (1) and (2)]. We derive the direct and inverse scattering problems for thev-dimensional radial Schrödinger equation for variable λ and fixed energy (negative or positive). We determine the scattering data (SD) in one to one correspondence with the potential and derive the corresponding Gelfand-Levitan equation. A family of exactly solvable potentials for any λ and fixed energy is obtained. The trace identities associated to this IST are derived for both signs of the energy. They relate integrals of local polynomials in the potential and its derivatives timesr 2n−1(n≧1) with the SD. The presence of this power ofr makes these relations useful in higher dimensions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
de Vega, H. J.: Phys. Lett. B.98B, 280 (1981)
de Vega, H. J.: LPTHE preprint (in preparation)
Regge, T.: Nuovo Cimento14, 951 (1959)
Burdet, G., Giffon, M., Predazzi, E.: Nuovo Cimento36, 1337 (1965)
Loeffeld, J. J.: Ann. Inst. Henri Poincaré8, 339 (1968)
Chadan, K., Sabatier, P. C.: Inverse Problems in Quantum Scattering Theory. Berlin, Heidelberg, New York: Springer 1977 (Specially chapters XI and XIII and references contained therein)
Naimark, M. A.: Linear Differential Operators, Part II. New York: Frederick Ingar 1968
Lebedev, N. N.: Special functions and their applications. London: Prentice Hall 1965
Erdélyi et al.: Tables of Integral Transforms, Vol. 2. New York: 1954 McGraw Hill
Newton, R. G.: Scattering Theory. New York: McGraw Hill 1966
Faddeev, L. D.: J. Math. Phys.4, 72 (1963)
See for example Buslaev, V. S., Faddeev, L. D.: Dokl. Akad. Nauk SSSR132, 13 (1960); Sov. Math. Dokl.1, 451 (1960)
Zakharov, V. E., Faddeev, L. D.: Funkt. Anal. iego Prilozh.5, 18 (1971) (Funct, Anal. Appl,5, 280 (1971))
Takhtadzhyan, L. A., Faddeev, L. D.: Teor. Mat. Fiz.21, 160 (1974) (Theor. Math. Phys. (USSR)21, 1046 (1975))
See also Faddeev, L. D.: In: Solitons, Topics in Current Physics. Bullough, R. M., Caudrey, P. J. (eds.) Berlin, Heidelberg, New York: Springer 1980
Ablowitz, M. J., Segur, H.: J. Math. Phys.16, 1054 (1975) I am indebted to Prof. Mark Ablowitz for calling my attention on this paper.
Birman, M. S., Krein, M. G.: Dokl. Akad. Nauk. SSSR144, 740 (1962)
Gelfand, I. M. Shilov, G. E.: Generalized Functions. New York: Academic Press 1964
Author information
Authors and Affiliations
Additional information
Communicated by E. Brézin
Laboratoire associé au CNRS
Rights and permissions
About this article
Cite this article
de Vega, H.J. The inverse scattering transformation in the angular momentum plane. Commun.Math. Phys. 81, 313–325 (1981). https://doi.org/10.1007/BF01209070
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01209070