Scattering in parabolic coordinates and a new representation for the scattering amplitude

Teodoro Azzarelli; Peter Collas

doi:10.1103/PhysRevD.12.3237

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Scattering in parabolic coordinates and a new representation for the scattering amplitude

Teodoro Azzarelli

Peter Collas

Xonics Inc., Van Nuys, California 91406

Department of Physics and Astronomy, California State University, Northridge, California 91324

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Phys. Rev. D 12, 3237 – Published 15 November, 1975

DOI: https://doi.org/10.1103/PhysRevD.12.3237

Abstract

We derive and investigate an exact integral representation of the scattering amplitude that results from the description of scattering in parabolic coordinates. The spectral function $a (s, ν)$ of this representation turns out to be an entire function of order 1 in the $ν$ variable provided that the partial-wave amplitude $A_{l} \sim l^{- \frac{1}{2}} e^{- l ξ}$ , $ξ > 0$ for $l$ large. We also briefly discuss the counterpart of the partial-wave series.

References (28)

N. N. Khuri, Phys. Rev. Lett. 10, 420 (1963) Phys. Rev. 132, 914 (1963)
[1]
N. N. Khuri, Lett. Nuovo Cimento 12, 601 (1975)
S. J. Wallace, Phys. Rev. D 9, 406 (1974) ibid.8, 1846 (1973) E. Predazzi, Ann. Phys. (N.Y.) 36, 228 (1966) ibid.36, 250 (1966) R. Blankenbecler and M. L. Goldberger, Phys. Rev. 126, 766 (1962) M. M. Islam, Saclay Report No. DPh-T/74/97, 1974 (unpublished)
Omitted endnote
H. Bateman, Tôhoku Math. J. 37, 23 (1933)
S. O. Rice, Duke Math. J. 6, 108 (1939)
W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966), p. 28
W. RudinHigher Transcendental Functions (Bateman Manuscript Project), edited by A. Erdélyi (McGraw-Hill, New York, 1953), Vol. 1, Sec. 3.8
Omitted endnote
E. Hille, Analytic Function Theory (Blaisdell, New York, 1959), Vol. 1, theorem 7.10.3, p. 194
E. J. Squires, Nuovo Cimento 25, 242 (1962)
A. O. Barut, The Theory of the Scattering Matrix (Macmillan, New York, 1967), p. 155
Omitted endnote
A. S. B. Holland, Introduction to the Theory of Entire Functions (Academic, New York, 1973)
Omitted endnote
H. Bateman, Ann. Math. 35, 767 (1934)
S. Pasternack, Philos. Mag. 28, 209 (1939)
Omitted endnote
Omitted endnote
L. J. Slater, Generalized Hypergeometric Functions (Cambridge Univ. Press, Cambridge, England, 1966), Eq. (2.3.3.7), p. 52
[9], Sec. 2.1
L. Carlitz, Can. J. Math. 9, 188 (1957)
J. Touchard, Can. J. Math. 8, 305 (1956)
H. Bateman, Duke Math. J. 4, 39 (1938)
H. Buchholz, The Confluent Hypergeometric Function (Springer, New York, 1969) [9]
L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Addison-Wesley, Reading, Mass., 1965), 2nd ed., p. 126
H. Buchholz, Math. Z. 52, 355 (1949)

[c1] N. N. Khuri, Phys. Rev. Lett. 10, 420 (1963) Phys. Rev. 132, 914 (1963)

[c2] [1]

[c3] N. N. Khuri, Lett. Nuovo Cimento 12, 601 (1975)

[c4] S. J. Wallace, Phys. Rev. D 9, 406 (1974) ibid.8, 1846 (1973) E. Predazzi, Ann. Phys. (N.Y.) 36, 228 (1966) ibid.36, 250 (1966) R. Blankenbecler and M. L. Goldberger, Phys. Rev. 126, 766 (1962) M. M. Islam, Saclay Report No. DPh-T/74/97, 1974 (unpublished)

[c5] Omitted endnote

[c6] H. Bateman, Tôhoku Math. J. 37, 23 (1933)

[c7] S. O. Rice, Duke Math. J. 6, 108 (1939)

[c8] W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1966), p. 28

[c9] W. RudinHigher Transcendental Functions (Bateman Manuscript Project), edited by A. Erdélyi (McGraw-Hill, New York, 1953), Vol. 1, Sec. 3.8

[c10] Omitted endnote

[c11] E. Hille, Analytic Function Theory (Blaisdell, New York, 1959), Vol. 1, theorem 7.10.3, p. 194

[c12] E. J. Squires, Nuovo Cimento 25, 242 (1962)

[c13] A. O. Barut, The Theory of the Scattering Matrix (Macmillan, New York, 1967), p. 155

[c14] Omitted endnote

[c15] A. S. B. Holland, Introduction to the Theory of Entire Functions (Academic, New York, 1973)

[c16] Omitted endnote

[c17] H. Bateman, Ann. Math. 35, 767 (1934)

[c18] S. Pasternack, Philos. Mag. 28, 209 (1939)

[c19] Omitted endnote

[c20] Omitted endnote

[c21] L. J. Slater, Generalized Hypergeometric Functions (Cambridge Univ. Press, Cambridge, England, 1966), Eq. (2.3.3.7), p. 52

[c22] [9], Sec. 2.1

[c23] L. Carlitz, Can. J. Math. 9, 188 (1957)

[c24] J. Touchard, Can. J. Math. 8, 305 (1956)

[c25] H. Bateman, Duke Math. J. 4, 39 (1938)

[c26] H. Buchholz, The Confluent Hypergeometric Function (Springer, New York, 1969) [9]

[c27] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Addison-Wesley, Reading, Mass., 1965), 2nd ed., p. 126

[c28] H. Buchholz, Math. Z. 52, 355 (1949)

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Scattering in parabolic coordinates and a new representation for the scattering amplitude

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