Prog. Theor. Phys. Vol. 12 No. 2 (1954) pp. 225-233
Upper and Lower Bounds of Born Approximation, I
Department of Physics, University of Tokyo
(Received April 30, 1954)
The properties of one dimensional Born approximation are investigated. It is shown that upper and lower bounds for any order Born approximation can be given by a single parameter (namely, by the convergence radius of Born expansion). In the convergency region of Born expansion it has been proved that, if the interaction force is (positive or negative) definite, the Born approximation of even order (including second) always gives a lower bound of the alsolute value of tan δ (δ: phase shift).
In appendix we also prove that, under the same conditions as above, the approximate method based on the Schwinger's variational principle which involves just the same computations as the second Born approximations, is always superior to the second Born approximation and gives a lower (upper) bound of the absolute value of tan δ.
DOI : 10.1143/PTP.12.225
- M. Born, Z. Physik. 38 (1926), 803.
N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions (Oxford University Press, London, 1949).
E. J. Williams, Rev. Mod. Phys. 17 (1945), 217[APS].
R. Jost and A. Pais, Phys. Rev. 82 (1951), 840[APS].
W. Kohn, Phys. Rev. 87 (1952), 539, [APS]and private communication to Prof. T. Kato.
Also see P. Urban and K. Wildermuth, Zeit. Naturforschg. a 8 (1953), 594.
For example, W. Kohn, Phys. Rev. 74 (1948), 1763[APS].
B. A. Lipmann and J. Schwinger, Phys. Rev. 79 (1950), 469[APS].
- T. Kato, Prog. Theor. Phys. 6 (1951), 394[PTP];
Phys. Rev. 80 (1950), 475[APS].
S. Altshuler, Phys. Rev. 89 (1953), 1278[APS].
G. F. Chew, Phys. Rev. 93 (1954), 341[APS].
In these papers the similar results have been observed by examples.
Citing Article(s) :
Progress of Theoretical Physics Vol. 14 No. 1 (1955) pp. 52-64
On the Meson-Theoretical Potentials
Shoichiro Otsuki and Ryozo Tamagaki