- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Reviews of Modern Physics

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Singular Potentials

WILLIAM M. FRANK, DAVID J. LAND, and RICHARD M. SPECTOR*

WILLIAM M. FRANK and DAVID J. LAND

  • U.S. Naval Ordnance Laboratory, Silver Spring, Maryland 20910

RICHARD M. SPECTOR*

  • Physics Department, Wayne State University, Detroit, Michigan 48202

  • *This author wishes to express his thanks to the U.S. Naval Ordnance Laboratory for its support and hospitality during the summer of 1968. Much of this article was written at that time at the Laboratory.
PDF

Rev. Mod. Phys. 43, 36 – Published 1 January, 1971

DOI: https://doi.org/10.1103/RevModPhys.43.36

Abstract

This article attempts to present a comprehensive review of the literature dealing with singular potentials through March 1969. The treatment is confined principally to nonrelativistic quantum mechanics, i.e., to solutions of the partial-wave Schrödinger equation with a singular potential. Some general physical and mathematical properties are given. Exact solutions are presented for those potentials for which they are available. Techniques which have been used in obtaining approximate solutions are outlined. Formal and physical applications of singular potentials are presented. Formal applications are those for which singular potentials have served as mathematical models illustrating concepts in elementary particle physics. These include applications to the Regge pole formalism, quantum field theories, and the peratization approximation. Physical applications entail those which have been made to molecular physics and to high-energy phenomenology.

View figure in article

References (164)

  1. M. A. Ahmed and D. B. Fairlie, Nuovo Cimento 38, 547 (1965)
  2. N. I. Akhieser and I. M. Glazman, M. NestellTheory of Linear Operators in Hilbert Space (translated from the Russian by (Ungar, New York, 1961)
  3. H. H. Aly, W. Guttinger, and H. J. W. Müller in Lectures on Particles and Fields, edited by H. H. Aly (Gordon and Breach, New York, 1970)
  4. H. H. AlyW. GuttingerH. J. W. Müller, and H. J. W. Müller, J. Math. Phys. 7, 1 (1966a)
  5. H. H. AlyW. GuttingerH. J. W. MüllerH. J. W. Müller, Nucl. Phys. 76, 241 (1966b)
  6. H. H. AlyW. GuttingerH. J. W. MüllerH. J. W. Müller, J. Math. Phys. 8, 367 (1967)
  7. H. H. AlyW. GuttingerH. J. W. MüllerH. J. W. Müller, and S. Okubo, Z. Physik 200, 20 (1967)
  8. H. H. AlyW. GuttingerH. J. W. MüllerH. J. W. MüllerS. Okubo, Riazuddin, and A. H. Zimerman, Phys. Rev. 136, B1174 (1964)
  9. H. H. AlyW. GuttingerH. J. W. MüllerH. J. W. MüllerS. OkuboRiazuddinA. H. Zimerman, Riazuddin, and A. H. Zimerman, Nuovo Cimento 35, 324 (1965b)
  10. H. H. AlyW. GuttingerH. J. W. MüllerH. J. W. MüllerS. OkuboRiazuddinA. H. ZimermanRiazuddinA. H. Zimerman, Riazuddin, and A. H. Zimerman, J. Math. Phys. 6, 1115 (1965a)
  11. H. H. AlyW. GuttingerH. J. W. MüllerH. J. W. MüllerS. OkuboRiazuddinA. H. ZimermanRiazuddinA. H. ZimermanRiazuddinA. H. Zimerman, and J. G. Taylor, J. Math. Phys. 9, 781 (1968)
  12. H. H. AlyW. GuttingerH. J. W. MüllerH. J. W. MüllerS. OkuboRiazuddinA. H. ZimermanRiazuddinA. H. ZimermanRiazuddinA. H. ZimermanJ. G. Taylor, and R. Wit, Z. f Physik 201, 129 (1967)
  13. I. Amdur and J. E. Jordan in Molecular Beams, edited by J. Ross (Wiley, New York, 1966), Chap. 2
  14. B. A. Arbuzov and A. T. Filippov, Phys. Letters 13, 95 (1964)
  15. B. A. ArbuzovA. T. Filippov, and A. T. Filippov, Nuovo Cimento 38, 796 (1965)
  16. B. A. ArbuzovA. T. FilippovA. T. Filippov, and A. T. Filippov, Zh. Eksp. Teor. Fiz. 49, 990 (1965) [Sov. Phys. JETP 22, 688 (1966)]
  17. B. A. ArbuzovA. T. FilippovA. T. FilippovA. T. Filippov, A. T. Filippov, and O. A. Khrustalev, Phys. Letters 8, 205 (1964)
  18. B. H. Armstrong and E. A. Power, Am. J. Phys. 31, 262 (1963)
  19. A. A. Arsenev, Dokl. Akad. Nauk SSSR 178, (1968) [Soviet Math. Dokl. 9, 253 (1968)]
  20. V. V. Babikov, Usp. Fiz. Nauk. 92, 3 (1967) [Soviet Physics Uspekhi 10, 271 (1967)]
  21. A. Bastai, L. Bertocchi, S. Fubini, G. Furlan, and M. Tonin, Nuovo Cimento 30, 1512 (1963a)
  22. A. BastaiL. BertocchiS. FubiniG. FurlanM. Tonin, L. Bertocchi, G. Furlan, and M. Tonin, Nuovo Cimento 30, 1532 (1963b)
  23. A. BastaiL. BertocchiS. FubiniG. FurlanM. ToninL. BertocchiG. FurlanM. ToninBateman Manuscript Project (McGraw-Hill, New York, 1953)
  24. H. Behncke, Nuovo Cimento 55, 780 (1968)
  25. R. B. Bernstein, in Molecular Beams, edited by J. Ross (Wiley, New York, 1966), Chap. 3
  26. L. Bertocchi, S. Fubini, and G. Furlan, Nuovo Cimento 32, 745 (1964b)
  27. L. BertocchiS. FubiniG. Furlan, S. Fubini, and G. Furlan, Nuovo Cimento 35, 599 (1965a)
  28. L. BertocchiS. FubiniG. FurlanS. FubiniG. Furlan, S. Fubini, and G. Furlan, Nuovo Cimento 35, 633 (1965b)
  29. L. BertocchiS. FubiniG. FurlanS. FubiniG. FurlanS. FubiniG. Furlan, M. McMillan, E. Predazzi, and M. Tonin, Nuovo Cimento 31, 1352 (1964a)
  30. J. E. Bowcock and A. P. Contogouris, Nuovo Cimento 33, 873 (1964)
  31. O. Brander, Phys. Letters 4, 218 (1963)
  32. F. Calogero, Nuovo Cimento 27, 261 (1963)
  33. F. Calogero, Phys. Rev. 135, B693 (1964)
  34. F. Calogero, Phys. Rev. 139, B602 (1965b)
  35. F. Calogero, Nuovo Cimento 37, 756 (1965b)
  36. F. Calogero, Variable Phase Approach to Potential Scattering (Academic, New York, 1967)
  37. F. Calogero, and M. Cassandro, Nuovo Cimento 34, 1712 (1964)
  38. F. CalogeroM. Cassandro, and M. Cassandro, Nuovo Cimento 37, 760 (1965)
  39. F. CalogeroM. CassandroM. Cassandro, and M. B. de Stefano, Phys. Rev. 146, 1195 (1966)
  40. D. S. Carter, Ph.D. thesis, Princeton, 1952
  41. K. M. Case, Phys. Rev. 80, 797 (1950)
  42. J. Challifour and R. J. Eden, J. Math. Phys. 4, 359 (1963)
  43. T. Y. Chang, Rev. Mod. Phys. 39, 911 (1967)
  44. J. M. Charap and N. Dombey, Phys. Letters 9, 210 (1964)
  45. J. M. CharapN. Dombey, and S. Fubini, Nuovo Cimento 14, 540 (1959)
  46. A. P. Contogouris, Nuovo Cimento 37, 1712 (1965)
  47. H. Cornille, Nuovo Cimento 33, 434 (1964)
  48. H. Cornille, Nuovo Cimento 36, 1316 (1965a)
  49. H. Cornille, Nuovo Cimento 38, 1243 (1965b)
  50. H. Cornille, Nuovo Cimento 39, 557 (1965c)
  51. H. Cornille, Nuovo Cimento 43A, 786 (1966)
  52. H. Cornille, G. Burdet, and M. Giffon, Nuovo Cimento 38, 647 (1965)
  53. H. CornilleG. BurdetM. Giffon, and E. Predazzi, Phys. Letters 10, 149 (1964)
  54. H. CornilleG. BurdetM. GiffonE. Predazzi, and E. Predazzi, Nuovo Cimento 35, 879 (1965a)
  55. H. CornilleG. BurdetM. GiffonE. PredazziE. Predazzi, and E. Predazzi, J. Math. Phys. 6, 1730 (1965b)
  56. H. CornilleG. BurdetM. GiffonE. PredazziE. PredazziE. PredazziG. Cosenza, L. Sertorio, and M. Toller, Nuovo Cimento 31, 1086 (1964)
  57. V. de Alfaro and E. Predazzi, Nuovo Cimento 39, 235 (1965)
  58. E. del Giudice and E. Galzenati, Nuovo Cimento 38, 443 (1965a)
  59. E. del GiudiceE. Galzenati, and E. Galzenati, Nuovo Cimento 40, 739 (1965b)
  60. N. Dombey, Nuovo Cimento 37, 1743 (1965)
  61. N. Dombey, and R. Jones, J. Math. Phys. 9, 986 (1968)
  62. M. A. Evgrafov, Asymptotic Estimates and Entire Functions (Gordon and Breach, New York, 1961)
  63. N. Faustov, Dokl. Akad. Nauk. SSSR 156, 1329 (1964) [Sov. Phys. Dokl. 9, 482 (1964)]
  64. G. Feinberg and A. Pais, Phys. Rev. 131, 2724 (1963)
  65. G. FeinbergA. Pais, and A. Pais, Phys. Rev. 133, B477 (1964)
  66. R. M. Felder, J. Chem. Phys. 49, 1438 (1968)
  67. A. T. Filippov, Phys. Letters 9, 78 (1964)
  68. W. M. Frank, J. Math. Phys. 8, 466 (1967)
  69. W. M. Frank, J. Math. Phys. 9, 1890 (1968)
  70. W. M. Frank, "Relative Jost Functions", Preprint a
  71. W. M. Frank, "The Phase Shift as a Bounded Functional," Preprint b
  72. W. M. Frank, "The Phase Shift as a Continuous Functional," II Preprint c
  73. W. M. Frank, and D. J. Land, J. Math. Phys. 11, 2041 (1970a)
  74. W. M. FrankD. J. Land, and D. J. Land, J. Math. Phys. 11, 2058 (1970b)
  75. W. M. FrankD. J. LandD. J. Land, and D. J. Land, "Some Inequalities for the Scattering Length for Repulsive Singular Potentials," Preprint c
  76. W. M. FrankD. J. LandD. J. LandD. J. Land, and D. J. Land, "Transitionally Singular Potentials," Preprint d
  77. N. Froman and P. O. Froman, JWKB Approximation: Contributions to the Theory (North Holland, Amsterdam, 1965)
  78. S. Fubini and R. Stroffolini, Nuovo Cimento 37, 1812 (1965)
  79. G. Furlan and G. M. Mahoux, Nuovo Cimento 36, 215 (1965)
  80. W. A. Gale, Ph.D. thesis, Rice University, Houston, Texas (1966a)
  81. W. A. Gale, J. Math. Phys. 7, 2171 (1966b)
  82. W. A. Gale, J. Math. Phys. 8, 1050 (1967)
  83. C. R. Garibotti, M. F. Pellicoro, and M. Villani, Nuovo Cimento 66A, 749 (1970)
  84. M. Giffon and E. Predazzi, Nuovo Cimento 33, 1374 (1964)
  85. J. S. Goldstein, Phys. Rev. 91, 1516 (1953)
  86. W. Gittinger, Nuovo Cimento 36, 969 (1965)
  87. W. Gittinger, R. Penzl, and E. Pfaffelhuber, Ann. Phys. (N.Y.) 33, 246 (1965a)
  88. W. GittingerR. PenzlE. Pfaffelhuber, R. Penzl, and E. Pfaffelhuber, Nuovo Cimento 35, 1247 (1965b)
  89. W. GittingerR. PenzlE. PfaffelhuberR. PenzlE. Pfaffelhuber, and E. Pfaffelhuber, Nuovo Cimento 43A, 423 (1966)
  90. M. B. Halpern, Ann. Phys. (N.Y.) 39, 351 (1966)
  91. M. B. HalpernHandbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun (National Bureau of Standards, 1964)
  92. R. A. Handelsman, Y. Pao, and J. S. Lew, Nuovo Cimento 55A, 453 (1968)
  93. B. Hartmann and Z. I. Slawsky, J. Chem. Phys. 47, 2491 (1967)
  94. J. Heading, Phase Integral Methods (Methuen, London, 1962)
  95. J. HeadingJ. O. Hirschfelder, Ed., Intermolecular Forces (Wiley, New York, 1967)
  96. J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954)
  97. R. J. Jabbur, Phys. Rev. 138, B1525 (1965)
  98. B. Jaksic and N. Limic, J. Math. Phys. 7, 88 (1966a)
  99. B. JaksicN. Limic, and N. Limic, Communications in Mathematical Physics 2, 94 (1966b)
  100. R. Jost, Helv. Phys. Acta 20, 256 (1947)
  101. N. N. Khuri and A. Pais, Rev. Mod. Phys. 36, 590 (1964)
  102. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1956), 2nd ed
  103. W. M. Kloet, and A. H. Zimerman, Nuovo Cimento 69A, 564 (1970)
  104. C. B. Kouris, Nuovo Cimento 44A, 598 (1966)
  105. D. J. Land, "Peratization of a Generalized Class of Exponentially Singular Potentials", Preprint, a
  106. L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, London, 1958) L. D. LandauE. M. LifshitzQuantum Mechanics (Pergamon, London, 1960)
  107. R. E. Langer, Phys. Rev. 51, 669 (1937)
  108. S. Y. Larsen, K. Witte, and J. E. Kilpatrick, J. Chem. Phys. 44, 213 (1966)
  109. T. D. Lee and C. N. Yang, Phys. Rev. 128, 885 (1962)
  110. T. D. LeeC. N. Yang, Phys. Rev. 128, 899 (1962)
  111. B. R. Levy and J. B. Keller, J. Math. Phys. 4, 54 (1963)
  112. N. Limic, Nuovo Cimento 26, 581 (1962)
  113. N. Limic, Nuovo Cimento 28, 1066 (1963)
  114. N. Limic, Nuovo Cimento 36, 100 (1965)
  115. A. A. Logunov and A. N. Tavkhelidze, Nuovo Cimento 29, 380 (1963a)
  116. A. A. Logunov, A. N. Tavkhelidze, I. T. Todorov, and O. A. Khrustalev, Nuovo Cimento 30, 134 (1963b)
  117. A. Martin, Suppl. Nuovo Cimento 21, 157 (1961)
  118. A. Martin, "Born Approximation and Dispersion Relations for Singular Potentials", Preludes in Theoretical Physics (Wiley, New York, 1966), p. 17
  119. D. Masson, J. Math. Phys. 8, 2308 (1967)
  120. N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford U. P., Oxford, 1947)
  121. K. Meetz, Nuovo Cimento 34, 690 (1964)
  122. J. Meixner and I. W. Schafke, Mathieusche Funktionen und Spharoid Funktionen (Springer, Berlin, 1954)
  123. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953)
  124. R. G. Newton, J. Math. Phys. 1, 319 (1960)
  125. R. G. Newton, Scattering Theory of Waves and Particles (McGraw-Hill, New York, 1966)
  126. T. F. O'Malley, L. Spruch, and L. Rosenberg, J. Math. Phys. 2, 491 (1961)
  127. A. Pais and T. T. Wu, J. Math. Phys. 5, 799 (1964a)
  128. A. PaisT. T. Wu, and T. T. Wu, Phys. Rev. 134, B1303 (1964b)
  129. A. Paliov and S. Rosendorff, J. Math. Phys. 8, 1829 (1967)
  130. H. Pauly and J. P. Toennies in Advances in Atomic and Molecular Physics, edited by D. R. Bates and I. Estermann (Academic, New York, 1965), Vol. I
  131. M. Plesset, Phys. Rev. 41, 278 (1932)
  132. E. Predazzi and T. Regge, Nuovo Cimento 24, 518 (1962)
  133. Y. Pwu and T. T. Wu, Phys. Rev. 133, B1299 (1964)
  134. M. E. Rose and R. R. Newton, Phys. Rev. 82, 470 (1951)
  135. R. F. Sawyer, Phys. Rev. 131, 1384 (1963)
  136. R. F. Sawyer, Phys. Rev. 135, B448 (1964)
  137. F. L. Scarf, Phys. Rev. 109, 2170 (1958)
  138. L. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1955) 2nd ed
  139. R. Serber, Phys. Rev. Letters 10, 357 (1963)
  140. R. Serber, Rev. Mod. Phys. 36, 649 (1964a)
  141. R. Serber, Phys. Rev. Letters 13, 32 (1964b)
  142. R. Spector, J. Math. Phys. 5, 1185 (1964)
  143. R. Spector, Nuovo Cimento 45, 924 (1966a)
  144. R. Spector, J. Math. Phys. 7, 2103 (1966b)
  145. R. Spector, J. Math. Phys. 8, 2357 (1967a)
  146. R. Spector, Bull. Am. Phys. Soc. 12, 50 (1967b)
  147. R. Spector, J. Math. Phys. 10, 503 (1969)
  148. R. Spector, and R. Chand, P. Theo. Phys. 39, 682 (1968)
  149. G. N. Stanciu, Nuovo Cimento 50A, 293 (1967)
  150. J. G. Taylor, Suppl. Nuovo Cimento 1, 857 (1963)
  151. R. M. Thaler, Phys. Rev. 114, 827 (1959)
  152. T. Tietz, Nuovo Cimento 11, 126 (1959)
  153. G. Tiktopoulos, Phys. Rev. 138, B1550 (1965)
  154. G. Tiktopoulos, and S. B. Treiman, Phys. Rev. 134, B844 (1964)
  155. P. B. Treacy, "A Singular Static Potential for α−α Elastic Scattering", Preprint
  156. G. Valiron, Theory of Integral Functions (Chelsea, New York, 1949)
  157. R. Vasudevan, K. Venkatesan, and G. Jagannathan, Nuovo Cimento Suppl. V, 621 (1967)
  158. E. Vogt and G. Wannier, Phys. Rev. 95, 1190 (1954)
  159. G. Wannier, Quart. Appl. Math. 11, 33 (1953)
  160. E. Whittaker and G. N. Watson, A Course of Modern Analysis, (Cambridge U. P., 1927), 4th ed
  161. A. S. Wightman, in Proceedings of the Fifth Annual Eastern Theoretical Physics Conference, edited by D. Feldman (Benjamin, New York, 1967)
  162. E. Wigner, Phys. Rev. 98, 145 (1955)
  163. T. T. Wu, Phys. Rev. 136, B1176 (1964)
  164. T. Y. Wu and T. Ohmura, Quantum Theory of Scattering (Prentice Hall, Englewood Cliffs, N.J., 1962)

Outline

Information

Rev. Mod. Phys. 43, 36– Published 1 January, 1971

    © 1971 American Physical Society

    Sign In to Your Journals Account

    Filter

    Filter

    Article Lookup

    Enter a citation