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Schrödinger Operators with Exactly Solvable Potentials

  • D. M. Gitman
  • I. V. Tyutin
  • B. L. Voronov
Chapter
Part of the Progress in Mathematical Physics book series (PMP, volume 62)

Abstract

There exist eleven families of the so-called exactly solvable potentials for which the one-dimensional Schrödinger equation can be solved exactly. In this Chapter we construct all the corresponding self-adjoint Schrödinger operators and perform their spectral analysis.

References

  1. 1.
    Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions. National Bureau of Standards, New York (1964)zbMATHGoogle Scholar
  2. 2.
    Adami, R., Teta, A.: On the Aharonov–Bohm effect. Lett. Math. Phys. 43, 43–54 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. Springer, Berlin (1988)zbMATHGoogle Scholar
  4. 4.
    Alford, M.G., March-Russel, J., Wilczek, F.: Enhanced baryon number violation due to cosmic strings. Nucl. Phys. B 328, 140–158 (1989)CrossRefGoogle Scholar
  5. 5.
    Alliluev, S.P.: The problem of collapse to the center in quantum mechanics. Sov. Phys. JETP 34, 8–13 (1972)Google Scholar
  6. 6.
    Aharonov, Y, Bohm, D.: Significance of electromagnetic potentials in quantum theory. Phys. Rev. 115, 485–491 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Akhiezer, N.I.: Lectures on Approximation Theory, 2nd edn. Nauka, Moscow (1963)Google Scholar
  8. 8.
    Akhiezer, A.I., Berestetskiǐ, V.B.: Quantum Electrodynamics. Interscience Publishers, New York (1965)Google Scholar
  9. 9.
    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Pitman, Boston (1981)zbMATHGoogle Scholar
  10. 10.
    Araujo, V.S., Coutinho, F.A.B., Perez, J.F.: On the most general boundary conditions for the Aharonov–Bohm scattering of a Dirac particle: helicity and Aharonov–Bohm symmetry conservation. J. Phys. A 34, 8859–8876 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Araujo, V.S., Coutinho, F.A.B., Perez, J.F.: Operator domains and self-adjoint operators. Amer. J. Phys. 72, 203–213 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Audretsch, J., Skarzinsky, V., Voronov, B.: Elastic scattering and bound states in the Aharonov–Bohm potential superinposed by an attractive ρ − 2 potential. J. Phys. A 34, 235–250 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Bagrov, V.G., Gitman, D.M.: Exact Solutions of Relativistic Wave Equations, Kluwer Acad. Publish., Dordrecht, Boston, London (1990)zbMATHGoogle Scholar
  14. 14.
    Bagrov, V.G., Gavrilov, S.P., Gitman, D.M., Meira Filho D.P.: Coherent states of non-relativistic electron in magnetic-solenoid field. J. Phys. A 43, 3540169 (2010); Coherent and semiclassical states in magnetic field in the presence of the Aharonov–Bohm solenoid. J. Phys. A: Math. Theor. 44, 055301 (2011)Google Scholar
  15. 15.
    Bagrov, V.G., Gitman, D.M., Tlyachev, V.B.: Solutions of relativistic wave equations in superpositions of Aharonov–Bohm, magnetic, and electric fields. J. Math. Phys. 42, 1933–1959 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Bagrov, V.G., Gitman, D.M., Levin, A., Tlyachev, V.B.: Impact of Aharonov–Bohm solenoid on particle radiation in magnetic field. Mod. Phys. Lett. A 16, 1171–1179 (2001)CrossRefGoogle Scholar
  17. 17.
    Bagrov, V.G., Gitman, D.M., Levin, A., Tlyachev, V.B.: Aharonov–Bohm effect in cyclotron and synchrotron radiations. Nucl. Phys. B 605, 425–454 (2001)zbMATHCrossRefGoogle Scholar
  18. 18.
    Bagrov, V.G., Gitman, D.M., Tlyachev, V.B.: l-Dependence of particle radiation in magnetic-solenoid field and Aharonov–Bohm effect. Int. J. Mod. Phys. A 17, 1045–1048 (2002)CrossRefGoogle Scholar
  19. 19.
    Ballhausen, C. J., Gajhede, M.: The tunnel effect and scattering by a negative Kratzer potential. Chem. Phys. Lett. 165(5) 449–452 (1990)CrossRefGoogle Scholar
  20. 20.
    Bateman, H., Erdélyi, A.: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953)Google Scholar
  21. 21.
    Bawin, M., Coon, S.A.: Singular inverse square potential, limit cycles, and self-adjoint extensions. Phys. Rev. A 67(5) 042712 (2003)CrossRefGoogle Scholar
  22. 22.
    Bayrak, O., Boztosun, I., Ciftci, H.: Exact analytical solutions to the Kratzer potential by the asymptotic iteration method. Int. J. Quantum Chem. 107, 540–544 (2007)CrossRefGoogle Scholar
  23. 23.
    le Bellac, M.: Quantum Physics. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  24. 24.
    Berezansky, Yu.M.: Eigenfunction Expansions Associated with Self-adjoint Operators. Naukova Dumka, Kiev (1965)Google Scholar
  25. 25.
    Berezin, F.A.: The Method of Second Quantization. Academic Press, New York (1966)zbMATHGoogle Scholar
  26. 26.
    Berezin, F.A., Faddeev, L.D.: A remark on Schrödinger’s equation with a singular potential. Sov. Math. Dokl. 2, 372–375 (1961)zbMATHGoogle Scholar
  27. 27.
    Berezin, F.A., Shubin, M.A.: Schrödinger Equation. Kluwer, New York (1991)zbMATHCrossRefGoogle Scholar
  28. 28.
    Bethe, H.A., Salpeter, E.E.: Quantum Mechanics of One- and Two-Electron Systems. Encyclopedia of Physics, vol. XXXV/1. Springer, Berlin (1957)Google Scholar
  29. 29.
    Billing, G.D., Adhikari, S.: The time-dependent discrete variable representation method in molecular dynamics. Chem. Phys. Lett. 321(3–4) 197–204 (2000)CrossRefGoogle Scholar
  30. 30.
    Bogoliubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields, 3rd edn. Wiley, New York (1980)Google Scholar
  31. 31.
    Bonneau, G., Faraut, J., Valent, G.: Self-adjoint extensions of operators and the teaching of quantum mechanics. Am. J. Phys. 69, 322–331 (2001)CrossRefGoogle Scholar
  32. 32.
    Bohm, D.: Quantum Theory. Prentice-Hall, Englewood Cliffs, NJ (1951)Google Scholar
  33. 33.
    Breitenecker, M., Grümm, H.-R.: Remarks on the paper by Bocchieri, P., Loinger, A.: “Nonexistence of the Aharonov–Bohm effect ”Nuovo Cim. A 55, 453–455 (1980)Google Scholar
  34. 34.
    Calogero, F.: Solution of a three-body problem in one dimension. J. Math. Phys. 10, 2191–2196 (1969)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Calogero, F.: Ground state of a one-dimensional N-body system. J. Math. Phys. 12, 2197–2200 (1969)CrossRefGoogle Scholar
  36. 36.
    Calogero, F.: Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials. J. Math. Phys. 12, 419–436 (1971)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Capri, A.: Nonrelativistic Quantum Mechanics. World Scientific Publishers, Singapore (2002)Google Scholar
  38. 38.
    Case, K.M.: Singular potentials. Phys. Rev. 80, 797–806 (1950)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Cohen-Tannoudji, C., Diu, B., Laloë, F.: Quantum Mechanics. Wiley, New York (1977)Google Scholar
  40. 40.
    Coutinho, F.A.B., Nogami, Y., Perez, J.F.: Self-adjoint extensions of the Hamiltonian for a charged-particle in the presence of a thread of magnetic-flux. Phys. Rev. A 46, 6052–6055 (1992); Self-adjoint extensions of the Hamiltonian for a charged spin-1/2 particle in the Aharonov–Bohm field. J. Phys. A 27, 6539–6550 (1994)Google Scholar
  41. 41.
    Coutinho, F.A.B., Perez, J.F.: Boundary-conditions in the Aharonov–Bohm scattering of Dirac particles and the effect of Coulomb interaction. Phys. Rev. D 48, 932–939 (1993)CrossRefGoogle Scholar
  42. 42.
    Coutinho, F.A.B., Perez, J.F.: Helicity conservation in the Aharonov–Bohm scattering of Dirac Particles. Phys. Rev. D 49, 2092–2097 (1994)CrossRefGoogle Scholar
  43. 43.
    Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators—with Applications to Quantum Mechanics and Global Geometry. Springer, Berlin (1987)Google Scholar
  44. 44.
    Davydov, A.S.: Quantum Mechanics, 2nd edn. Pergamon Press, Oxford/New York (1976)Google Scholar
  45. 45.
    Dirac, P.A.M.: The Quantum Theory of the Electron. Proc. Roy. Soc. Lond., A 117, 610–624 (1928); The Quantum Theory of the Electron. Part II, Proc. Roy. Soc. Lond., A 118, 351–361 (1928); Darwin, C.G.: The Wave Equation of the Electron. Proc. Roy. Soc. Lond., A 118, 654–680 (1928); Gordon, W.: Die Energieniveaus des Wasserstoffatoms nach der Dirackschen Quanten Theorie des Electrons. Zs. Phys. 48, 11–15 (1928); Gordon, E.U., Shortley, G.H.: The Theory of Atomic Spectra. Cambridge University Press, Cambridge (1935)Google Scholar
  46. 46.
    Dirac, P.A.M.: The theory of magnetic poles. Phys. Rev. 74, 817–830 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Dirac, P.A.M.: Quantized singularities in the electromagnetic field. Proc. Royal Soc. (London) A 133, 60–72 (1931)Google Scholar
  48. 48.
    Dirac, P.A.M.: The Principles of Quantum Mechanics. Clarendon Press, Oxford (1958)zbMATHGoogle Scholar
  49. 49.
    Dirac, P.A.M.: Lectures on Quantum Mechanics. Belfer Graduate School of Science, Yeshiva University, New York (1964)Google Scholar
  50. 50.
    Ditkin, V.A., Prudnikov A.P.: Integral Transformations and Operational Calculus. FizMatLit, Moscow (1961)Google Scholar
  51. 51.
    Dunford, N., Schwartz, J.T.: Linear operators, part II. Spectral theory. Self adjoint operators in Hilbert space. Interscience Publishers, New York (1963)zbMATHGoogle Scholar
  52. 52.
    Eckart, C.: The penetration of a potential barrier by electrons. Phys. Rev. 35, 1303–1309 (1930)CrossRefGoogle Scholar
  53. 53.
    Ehrenberg, W, Siday, R.E.: The refractive index in electron optics and the principles of dynamics. Proc. Phys. Soc. Lond., B 62, 8–21 (1949)Google Scholar
  54. 54.
    Eliashevich, M.A.: Atomic and Molecular Spectroscopy. State Physical and Mathematical Publishing, Moscow (1962)Google Scholar
  55. 55.
    Exner, P., Št’oviček, P., Vytřas, P.: Generalized boundary conditions for the Aharonov–Bohm effect combined with a homogeneous magnetic field. J. Math. Phys. 43, 2151–2168 (2002)Google Scholar
  56. 56.
    Faddeev, L.D., Maslov, B.P.: Operators in Quantum Mechanics. In: Krein, S.G. (ed.) Spravochnaya Matematicheskaya Biblioteka (Functional Analysis). Nauka, Moscow (1964)Google Scholar
  57. 57.
    Faddeev, L.D., Yakubovsky, O.A.: Lectures on Qunatum Mechanics. Leningrad State University Press, Leningrad (1980)Google Scholar
  58. 58.
    Flekkøy, E.G., Leinaas, J.M.: Vacuum currents around a magnetic fluxstring. Int. J. Mod. Phys. A 6, 5327–5347 (1991)CrossRefGoogle Scholar
  59. 59.
    Flügge, S.: Practical Quantum Mechanics, vol I. Springer, Berlin (1994)Google Scholar
  60. 60.
    Fradkin, E.S., Gitman, D.M., Shvartsman, Sh.M.: Quantum Electrodynamics with Unstable Vacuum. Springer, Berlin (1991)Google Scholar
  61. 61.
    Fues, E.: Das Eigenschwingungs spektrum zweiatomiger molekule in der Undulationsmechanik. Ann. Phys. 80, 376–396 (1926)Google Scholar
  62. 62.
    Furry, W.H.: On bound states and scattering in positron theory. Phys. Rev. 81, 115–124 (1951)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Galindo, A., Pascual, P.: Quantum Mechanics, vols. 1 and 2. Springer (1990, 1991)Google Scholar
  64. 64.
    Gasiorowicz, S.: Quantum Physics. Wiley, New York (1974)Google Scholar
  65. 65.
    Gavrilov, S.P., Gitman, D.M.: Quantization of point-like particles and consisitent relativistic quantum mechanics. Int. J. Mod. Phys. A 15, 4499–4538 (2000)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Gavrilov, S.P., Gitman, D.M., Smirnov, A.A.: Dirac equation in the magnetic-solenoid field. Euro. Phys. J. C 30, 009 (2003); 32(Suppl.) 119–142 (2003)Google Scholar
  67. 67.
    Gavrilov, S.P., Gitman, D.M., Smirnov, A.A.: Self-adjoint extensions of Dirac Hamiltonian in magnetic-solenoid field and related exact solutions. Phys. Rev. A 67(4) 024103 (2003)CrossRefGoogle Scholar
  68. 68.
    Gavrilov, S.P., Gitman, D.M., Smirnov, A.A.: Green functions of the Dirac equation with magnetic-solenoid field. J. Math. Phys. 45, 1873–1886 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Gavrilov, S.P., Gitman, D.M., Smirnov, A.A., Voronov, B.L.: Dirac fermions in a magnetic-solenoid field. In: Benton, C.V. (ed.) Focus on Mathematical Physics Research, pp. 131–168. Nova Science Publishers, New York (2004)Google Scholar
  70. 70.
    Gelfand, I.M., Kostyuchenko, A.G.: Eigenfunction expansions for differential and other operators. Dokl. Akad. Nauk SSSR 103(3) 349–352 (1955)MathSciNetGoogle Scholar
  71. 71.
    Gelfand, I.M., Shilov, G.E.: Some problems of the theory of differential equations. Generalized functions, part 3. Fizmatgiz, Moscow (1958)Google Scholar
  72. 72.
    Gerbert, Ph. de S., Jackiw, R.: Classical and quantum scattering on a spinning cone. Commun. Math. Phys. 124, 229–260 (1989)Google Scholar
  73. 73.
    Gerbert, Ph. de S.: Fermions in an Aharonov–Bohm field and cosmic strings. Phys. Rev. D 40, 1346–1349 (1989)Google Scholar
  74. 74.
    Gieres, F.: Mathematical surprises and Dirac’s formalism in quantum mechanics. Rep. Prog. Phys. 63, 1893–1931 (2000)CrossRefGoogle Scholar
  75. 75.
    Gitman, D.M., Tyutin, I.V.: Quantization of Fields with Constraints. Springer, Berlin (1990)zbMATHCrossRefGoogle Scholar
  76. 76.
    Gitman, D.M., Tyutin, I.V., Voronov, B.L.: Self-adjoint extensions and spectral analysis in Calogero problem. J. Phys. A 43, 145205 (2010)MathSciNetCrossRefGoogle Scholar
  77. 77.
    Gitman, D.M., Tyutin, I.V., Voronov, B.L.: Oscillator representations for self-adjoint Calogero Hamiltonians. Journ. Phys. A Math. Theor. 44 425204 (2011)MathSciNetCrossRefGoogle Scholar
  78. 78.
    Gitman, D.M., Tyutin, I.V., Smirnov, A., Voronov, B.L.: Self-adjoint Schrödinger and Dirac operators with Aharonov–Bohm and magnetic-solenoid fields. Phys. Scr. 85 (2012) 045003CrossRefGoogle Scholar
  79. 79.
    Gorbachuk, V.I., Gorbachuk, M.L., Kochubei, A.N.: Extension theory for symmetric operators and boundary value problems for differential equations. Ukr. Math. J. 41(10) 1117–1129 (1989); translation from Ukr. Mat. Zh. 41(10) 1299–1313 (1989)Google Scholar
  80. 80.
    Gorbachuk, V.I., Gorbachuk, M.L.: Boundary Value Problems for Operator Differential Equations. Kluwer, Dordrecht (1991)Google Scholar
  81. 81.
    Gradshtein, I.S., Ryzhik, I.W.: Table of Integrals, Series, and Products. Academic Press, New York (1994)Google Scholar
  82. 82.
    Greiner, W., Müller, B., Rafelski, J.: Quantum Electrodynamics of Strong Fields. Springer, Berlin (1985)Google Scholar
  83. 83.
    Gustafson, S.J., Sigal, I.M.: Mathematical Concepts of Quantum Mechanics. Universitext. Springer, Berlin (2003)zbMATHCrossRefGoogle Scholar
  84. 84.
    Haag, R.: Local Quantum Physics. Springer, Berlin (1996)zbMATHGoogle Scholar
  85. 85.
    Hagen, C.R.: Aharonov–Bohm scattering of particles with spin. Phys. Rev. Lett. 64, 503–506 (1990); Spin dependence of the Aharonov–Bohm effect. Int. J. Mod. Phys. A 6, 3119–3149 (1991)Google Scholar
  86. 86.
    Hagen, C.R.: Effects of nongauge potentials on the spin-1/2 Aharonov–Bohm problem. Phys. Rev. D 48, 5935–5939 (1993)CrossRefGoogle Scholar
  87. 87.
    Halmos, P.R.: The Hilbert space problem book. D. van Nostrand Co., Inc. Toronto, London (1967)Google Scholar
  88. 88.
    Halperin, I.: Introduction to the Theory of Distributions. University of Toronto Press, Toronto (1952) (Based on the lectures given by Laurent Schwartz)zbMATHGoogle Scholar
  89. 89.
    Hamilton, J.: Aharonov–Bohm and Other Cyclic Phenomena. Springer Tracts in Modern Physics. Springer, New York (1997)zbMATHGoogle Scholar
  90. 90.
    Hartman, P., Wintner, A.: Criteria of non-degeneracy for the wave equations. Am. J. Math. 70, 295–269 (1948)MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, Princeton (1992)zbMATHGoogle Scholar
  92. 92.
    Hislop, P.D., Sigal, I.M.: Introduction to Spectral Theory: With Applications to Schrödinger Operators. Appl. Math. Sci. Springer (1995)Google Scholar
  93. 93.
    Hutson, V.C.L., Pym, J.S.: Applications of Functiomal Analysis and Operator Theory. Academic Press, London (1980)Google Scholar
  94. 94.
    Jackiw, R.: Delta function potentials in two- and three-dimensional quantum mechanics. In: Ali, A, Hoodbhoy, P. (eds.) M.A.B. Bèg Memorial Volume. World Scientific, Singapore (1991)Google Scholar
  95. 95.
    Jörgens, K., Weidmann, J.: Spectral Properties of Hamiltonian Operators. Lecture Notes in Mathematics. Springer, Berlin (1973)zbMATHGoogle Scholar
  96. 96.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)zbMATHGoogle Scholar
  97. 97.
    Kolmogorov, A.N., Fomin, S.V.: Elements of Function Theory and Functional Analysis. Nauka, Moskva (1976)Google Scholar
  98. 98.
    Konishi, K., Paffuti, G.: Quantum Mechanics: A New Introduction. Oxford University Press, Oxford (2009)zbMATHGoogle Scholar
  99. 99.
    Kostuchenko, A.G., Krein, S.G., Sobolev, V.I.: Linear Operators in Hilbert Space. In: Krein, S.G. (ed.) Spravochnaya Matematicheskaya Biblioteka (Functional Analysis). Nauka, Moscow (1964)Google Scholar
  100. 100.
    Kratzer, A.: Die Ultraroten Rotationsspektren der Halogenwasserstoffe. Z. Phys. 3(5) 289–307 (1920);M.C. Baldiotti, D.M. Gitman, I.V. Tyutin, and B.L. Voronov, Self-adjoint extensions and spectral analysis in the generalized Kratzer problem, Phys. Scr. 83 (2011) 065007Google Scholar
  101. 101.
    Krein, M.T.: A general method for decomposition of positively defined kernels into elementary products. Dokl. Akad. Nauk SSSR 53, 3–6 (1946) (in Russian); On Hermitian operators with guiding functionals. Zbirnik Prazc’ Institutu Matematiki, AN URSR No.10 83–105 (1948) (in Ukranian)Google Scholar
  102. 102.
    Kuzhel, A.V., Kuzhel, S.A.: Regular Extensions of Hermitian Operators. VSP, Utrecht (1998)zbMATHGoogle Scholar
  103. 103.
    Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Pergamon Press, Oxford (1975)Google Scholar
  104. 104.
    Landau, L.D., Lifshitz, E.M.: Quantum Mechanics: Non-Relativistic Theory. Pergamon Press, Oxford (1977)Google Scholar
  105. 105.
    Lemus, R., Bernal, R.: Connection of the vibron model with the modified Pöschl–Teller potential in configuration space. Chem. Phys. 283(3) 401–417 (2002)CrossRefGoogle Scholar
  106. 106.
    Levinson, N.: Criteria for the limit point case fir second order linear differential operators. Casopis Pěst. Math. Fys. 74, 17–20 (1949)MathSciNetGoogle Scholar
  107. 107.
    Lewis, R.R.: Aharonov–Bohm effect for trapped ions. Phys. Rev. A 28, 1228–1236 (1983)CrossRefGoogle Scholar
  108. 108.
    Levitan, B.M.: Eigenfunction Expansions Assosiated with Second-order Differential Equations. Gostechizdat, Moscow (1950) (in Russian)Google Scholar
  109. 109.
    Liboff, R.L.: Introduction to Quantum Mechanics. Addison-Wesley, New York (1994)Google Scholar
  110. 110.
    Lisovyy, O.: Aharonov–Bohm effect on the Poincaré disk. J. Math. Phys. 48, 052112-17 (2007). doi:10.1063/1.2738751MathSciNetCrossRefGoogle Scholar
  111. 111.
    Meetz, K.: Singular Potentials in Nonrelativistic Quantum Mechanics. IL Nuovo Cimento 34, 690–708 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  112. 112.
    Messiah, A.: Quantum Mechanics. Interscience, New York (1961)Google Scholar
  113. 113.
    Morse, P.M.: Diatomic molecules according to the wave mechanics. II. Vibrational levels. Phys. Rev. 34, 57–64 (1929)Google Scholar
  114. 114.
    Morse, P.M., Fisk, J.B., Schiff, L.I.: Collision of neutron and proton. Phys. Rev. 50, 748–754 (1936)CrossRefGoogle Scholar
  115. 115.
    Mott, N.F., Massey, H.S.W.: Theory of Atomic Collisions. Oxford University Press, Oxford (1933)zbMATHGoogle Scholar
  116. 116.
    Naimark, M.A.: Linear differential operators. Nauka, Moskva (1959) (in Russian). F. Ungar Pub. Co. New York (1967)Google Scholar
  117. 117.
    Nambu, Y.: The Aharonov–Bohm problem revisited. Nucl. Phys. B 579, 590–616 (2000); Hirokawa, M., Ogurisu, O.: Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field. J. Math. Phys. 42, 3334–3343 (2001)Google Scholar
  118. 118.
    Narnhofer, H.: Quantum theory for 1 ∕ r 2 potentials. Acta Phys. Aust. 40, 306–322 (1974)MathSciNetGoogle Scholar
  119. 119.
    Nikishov, A.I.: The role of connection between spin and statistics in QED with pair creating external field. In: Problems in Theoretical Physics. Collection in commemoration of I.E. Tamm, pp. 299–305. Nauka, Moscow (1972); Problems of Intensive External Fields in Quantum Electrodynamics. In: Quantum Electrodynamics of Phenomena in Intense Fields, Proc. P.N. Lebedev Phys. Inst., 111, pp. 153–271. Nauka, Moscow (1979); Bagrov, V.G., Gitman, D.M., Shvartsman, Sh.M.: Concerning the production of electron–positron pairs from vacuum. Sov. Phys. JETP 41, 191–194 (1975)Google Scholar
  120. 120.
    Olariu, S., Popescu, I.I.: The quantum effects of electromagnetic fluxes. Rev. Mod. Phys. 57, 339–436 (1985)CrossRefGoogle Scholar
  121. 121.
    Oliveira C.R. de, Pereira, M.: Mathematical justification of the Aharonov–Bohm Hamiltonian. J. Stat. Phys. 133, 1175–1184 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  122. 122.
    Pomeranchuk I., Smorodinsky, Ya.: On energy levels in systems with Z > 137. J. Phys. (USSR) 9, 97–100 (1945); Gershtein S.S., Zel’dovich, Ya.B.: Positron production during the mutual approach of heavy nuclei and the polarization of the vacuum. Sov. Phys. JETP 30, 358–361 (1970)Google Scholar
  123. 123.
    Perelomov A.M., Popov, V.S.: Fall to the center in quantum mechanics. Theor. Math. Phys. 4, 664–677 (1970)CrossRefGoogle Scholar
  124. 124.
    Peshkin M., Tonomura, A.: The Aharonov–Bohm Effect. Lecture Notes in Physics. Springer, New York (1989)CrossRefGoogle Scholar
  125. 125.
    Plesner, A.I.: Spectral Theory of Linear Operators. Nauka, Moscow (1965)Google Scholar
  126. 126.
    Pöschl, G., Teller, E.: Bemerkungen zur Quantenmechanik des Anharmonischen Oszillators. Z. Phys. 83(3–4) 143–151 (1933)zbMATHGoogle Scholar
  127. 127.
    Putnam, C.R.: On the spectra of certain boundary value problem. Am. J. Math. 71, 109–111 (1948)MathSciNetCrossRefGoogle Scholar
  128. 128.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I. Functional Analysis. Academic Press, New York (1980)zbMATHGoogle Scholar
  129. 129.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. II. Harmonic Analysis. Self-adjointness. Academic Press, New York (1975)zbMATHGoogle Scholar
  130. 130.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. IV. Analysis of Operators. Academic Press, New York (1978)zbMATHGoogle Scholar
  131. 131.
    Richtmyer, R.D.: Principles of Advanced Mathematical Physics, vol. 1. Springer, New York (1978)zbMATHGoogle Scholar
  132. 132.
    Riesz, F., Sz.-Nagy, B.: Lecons d’Analyse Fonctionnelle. Akademiai Kiado, Budapest (1972)Google Scholar
  133. 133.
    Rose, M.E.: Relativistic Electron Theory. Wiley, New York (1961)zbMATHGoogle Scholar
  134. 134.
    Rosen, N., Morse, P.M.: On the vibrations of polyatomic molecules. Phys. Rev. 42, 210–215 (1932)zbMATHCrossRefGoogle Scholar
  135. 135.
    Ruijsenaars, S.N.M.: The Aharonov–Bohm effect and scattering theory. Ann. Phys. 146, 1–34 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  136. 136.
    Sakurai, J.J.: Modern Quantum Mechanics. Addison-Wesley, New York (1994)Google Scholar
  137. 137.
    Scarf, S.L.: Discrete states for singular potential problems. Phys. Rev. 109, 2170–2176 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  138. 138.
    Schiff, L.I.: Quantum Mechanics. McGraw–Hill, New York (1955)Google Scholar
  139. 139.
    Schweber, S.: An Introduction to Relativistic Quantum Field Theory. Harper & Row, New York (1961)Google Scholar
  140. 140.
    Shilov, G.E.: Mathematical Analysis. Second special course. Nauka, Moscow (1965)zbMATHGoogle Scholar
  141. 141.
    Shilov, G.E., Gurevich, B.L.: Integral, measure, and derivative. Nauka, Moscow (1967)Google Scholar
  142. 142.
    Stone, M.H.: Linear Transformations in Hilbert space and their applications to analysis. Am. Math. Soc., vol. 15. Colloquium Publications, New York (1932)Google Scholar
  143. 143.
    Stepanov, V.V.: Course of differential equations. GIFML, Moskva (1959)Google Scholar
  144. 144.
    Takhtajan, L.A.: Quantum Mechanics for Mathematicians. Graduate Studies in Mathematics, 95. American Mathematical Society (2008)Google Scholar
  145. 145.
    Teschl, G.: Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators. Graduate Studies in Mathematics, 99. American Mathematical Society (2009)Google Scholar
  146. 146.
    Thaller, B.: The Dirac Equation, Texts and Monographs in Physics. Springer, Berlin (1992)Google Scholar
  147. 147.
    Thirring, W.: Quantum Mathematical Physics – Atoms, Molecules and Large Systems. Springer, Berlin (2002)Google Scholar
  148. 148.
    Titchmarsh, E.C.: Eigenfunction Expansions Assosiated with Second-order Differential Equations. Clarendon Press, Oxford (1946)Google Scholar
  149. 149.
    Titchmarsh, E.C.: Eigenfunction Expansions Assosiated with Second-order Differential Equations. Part II. Clarendon Press, Oxford (1958)Google Scholar
  150. 150.
    Tyutin, I.V.: Electron scattering on a solenoid. Preprint FIAN (P.N. Lebedev Physical Institute, Moscow) no. 27. arXiv:0801.2167 (quant-ph) (1974)Google Scholar
  151. 151.
    van Haeringen, H.: Bound states for r  − 2 potentials in one and three dimensions. J. Math. Phys. 19, 2171–2179 (1978)MathSciNetCrossRefGoogle Scholar
  152. 152.
    Villalba, V.M.: Exact solutions of the Dirac equation for a Coulomb and scalar potential in the presence of an Aharonov–Bohm and magnetic monopole fields. J. Math. Phys. 36, 3332–3344 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  153. 153.
    von Neumann, J.: Mathematische Grundlagen der Quantenmechanik. Springer, Berlin (1932)zbMATHGoogle Scholar
  154. 154.
    von Neumann, J.: Functional operators. The Geometry of Orthogonal Spaces, vol. 2. Princeton University Press, Princeton (1950)Google Scholar
  155. 155.
    Voronov, B.L., Gitman, D.M., Tyutin, I.V.: The Dirac Hamiltonian with a superstrong Coulomb field. Theor. Math. Phys. 150(1) 34–72 (2007);D.M. Gitman, A.D. Levin, I.V. Tyutin, B.L. Voronov, Electronic Structure of Superheavy Atoms. Revisited, arXiv:1112.2648, quant-ph (2012)Google Scholar
  156. 156.
    Voronov, B.L., Gitman, D.M., Tyutin, I.V.: Constructing quantum observables and self-adjoint extensions of symmetric operators.I. Russ. Phys. J. 50(1) 1–31 (2007)Google Scholar
  157. 157.
    B.L. Voronov, D.M. Gitman, I.V.Tyutin, Constructing quantum observables and self-adjoint extensions of symmetric operators. II. Differential operators, Russ. Phys. J. 50/9 853–884 (2007)Google Scholar
  158. 158.
    Voronov, B.L., Gitman, D.M., Tyutin, I.V.: Constructing quantum observables and self-adjoint extensions of symmetric operators. III. Self-adjoint boundary conditions. Russ. Phys. J. 51(2) 115–157 (2008)MathSciNetzbMATHGoogle Scholar
  159. 159.
    Voropaev, S.A., Galtsov, D.V., Spasov, D.A.: Bound states for fermions in the gauge Aharonov–Bohm field. Phys. Lett. B 267, 91–94 (1991)MathSciNetCrossRefGoogle Scholar
  160. 160.
    Weidmann, J.: Spectral Theory of Ordinary Differential Operators. Springer, Berlin (1987)zbMATHGoogle Scholar
  161. 161.
    Weyl, H.: Über Gewöhnliche Lineare Differentialgleichungen mit Singulären Stellen und ihre Eigenfunctionen, pp. 37–64. Göttinger Nachrichten (1909)Google Scholar
  162. 162.
    Weyl, H.: Über Gewöhnliche differentialgleichungen mit singularitäten und zugehörigen entwicklungen willkürlicher funktionen. Math. Annal. 68, 220–269 (1910)MathSciNetzbMATHCrossRefGoogle Scholar
  163. 163.
    Weyl, H.: Über Gewöhnliche Differentialgleichungen mit Singulären Stellen und ihre Eigenfunctionen, pp. 442–467. Göttinger Nachrichten (1910)Google Scholar
  164. 164.
    Whittaker E.T., Watson, G.N.: A Course of Modern Analysis, vol. 2. Cambridge University Press, Cambridge (1927)zbMATHGoogle Scholar
  165. 165.
    Wu, T.T., Yang, C.N.: Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Rev. D 12, 3845–3857 (1975)MathSciNetCrossRefGoogle Scholar
  166. 166.
    Zel’dovich, Ya. B., Popov, V.S.: Electronic Structure of Superheavy Atoms. Sov. Phys. Uspekhi 14, 673–694 (1972)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • D. M. Gitman
    • 1
  • I. V. Tyutin
    • 2
  • B. L. Voronov
    • 2
  1. 1.Instituto de FísicaUniversidade de São PauloSão PauloBrasil
  2. 2.Department of Theoretical PhysicsP.N. Lebedev Physical InstituteMoscowRussia

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