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Differential Operators

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

Abstract

The problem of constructing self-adjoint ordinary differential operators starting from self-adjoint differential operations is discussed based on the general theory of self-adjoint extensions of symmetric operators outlined in the previous Chap. 4. We describe various methods for specifying self-adjoint operators associated with self-adjoint differential operations by boundary conditions. An attention is focused on features peculiar to differential operators, among them a notion of natural domain and a representation of asymmetry forms of the adjoint operator in terms of boundary forms.

Keywords

Differential Operator Symmetric Operator Finite Interval Boundary Form Left Endpoint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Abramowitz, M., Stegun, I. (eds.): Handbook of Mathematical Functions. National Bureau of Standards, New York (1964)MATHGoogle Scholar
  2. 2.
    Adami, R., Teta, A.: On the Aharonov–Bohm effect. Lett. Math. Phys. 43, 43–54 (1998)MathSciNetMATHCrossRefGoogle Scholar
  3. 5.
    Alliluev, S.P.: The problem of collapse to the center in quantum mechanics. Sov. Phys. JETP 34, 8–13 (1972)Google Scholar
  4. 9.
    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Pitman, Boston (1981)MATHGoogle Scholar
  5. 11.
    Araujo, V.S., Coutinho, F.A.B., Perez, J.F.: Operator domains and self-adjoint operators. Amer. J. Phys. 72, 203–213 (2004)MathSciNetMATHCrossRefGoogle Scholar
  6. 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
  7. 24.
    Berezansky, Yu.M.: Eigenfunction Expansions Associated with Self-adjoint Operators. Naukova Dumka, Kiev (1965)Google Scholar
  8. 27.
    Berezin, F.A., Shubin, M.A.: Schrödinger Equation. Kluwer, New York (1991)MATHCrossRefGoogle Scholar
  9. 51.
    Dunford, N., Schwartz, J.T.: Linear operators, part II. Spectral theory. Self adjoint operators in Hilbert space. Interscience Publishers, New York (1963)Google Scholar
  10. 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
  11. 71.
    Gelfand, I.M., Shilov, G.E.: Some problems of the theory of differential equations. Generalized functions, part 3. Fizmatgiz, Moscow (1958)Google Scholar
  12. 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
  13. 80.
    Gorbachuk, V.I., Gorbachuk, M.L.: Boundary Value Problems for Operator Differential Equations. Kluwer, Dordrecht (1991)Google Scholar
  14. 88.
    Halperin, I.: Introduction to the Theory of Distributions. University of Toronto Press, Toronto (1952) (Based on the lectures given by Laurent Schwartz)MATHGoogle Scholar
  15. 93.
    Hutson, V.C.L., Pym, J.S.: Applications of Functiomal Analysis and Operator Theory. Academic Press, London (1980)Google Scholar
  16. 96.
    Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)MATHGoogle Scholar
  17. 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
  18. 102.
    Kuzhel, A.V., Kuzhel, S.A.: Regular Extensions of Hermitian Operators. VSP, Utrecht (1998)MATHGoogle Scholar
  19. 108.
    Levitan, B.M.: Eigenfunction Expansions Assosiated with Second-order Differential Equations. Gostechizdat, Moscow (1950) (in Russian)Google Scholar
  20. 116.
    Naimark, M.A.: Linear differential operators. Nauka, Moskva (1959) (in Russian). F. Ungar Pub. Co. New York (1967)Google Scholar
  21. 118.
    Narnhofer, H.: Quantum theory for 1 ∕ r 2 potentials. Acta Phys. Aust. 40, 306–322 (1974)Google Scholar
  22. 123.
    Perelomov A.M., Popov, V.S.: Fall to the center in quantum mechanics. Theor. Math. Phys. 4, 664–677 (1970)CrossRefGoogle Scholar
  23. 125.
    Plesner, A.I.: Spectral Theory of Linear Operators. Nauka, Moscow (1965)Google Scholar
  24. 128.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. I. Functional Analysis. Academic Press, New York (1980)Google Scholar
  25. 130.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, vol. IV. Analysis of Operators. Academic Press, New York (1978)Google Scholar
  26. 131.
    Richtmyer, R.D.: Principles of Advanced Mathematical Physics, vol. 1. Springer, New York (1978)MATHGoogle Scholar
  27. 132.
    Riesz, F., Sz.-Nagy, B.: Lecons d’Analyse Fonctionnelle. Akademiai Kiado, Budapest (1972)Google Scholar
  28. 140.
    Shilov, G.E.: Mathematical Analysis. Second special course. Nauka, Moscow (1965)MATHGoogle Scholar
  29. 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
  30. 143.
    Stepanov, V.V.: Course of differential equations. GIFML, Moskva (1959)Google Scholar
  31. 148.
    Titchmarsh, E.C.: Eigenfunction Expansions Assosiated with Second-order Differential Equations. Clarendon Press, Oxford (1946)Google Scholar
  32. 149.
    Titchmarsh, E.C.: Eigenfunction Expansions Assosiated with Second-order Differential Equations. Part II. Clarendon Press, Oxford (1958)Google Scholar
  33. 151.
    van Haeringen, H.: Bound states for r  − 2 potentials in one and three dimensions. J. Math. Phys. 19, 2171–2179 (1978)Google Scholar
  34. 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
  35. 160.
    Weidmann, J.: Spectral Theory of Ordinary Differential Operators. Springer, Berlin (1987)MATHGoogle Scholar
  36. 161.
    Weyl, H.: Über Gewöhnliche Lineare Differentialgleichungen mit Singulären Stellen und ihre Eigenfunctionen, pp. 37–64. Göttinger Nachrichten (1909)Google Scholar
  37. 162.
    Weyl, H.: Über Gewöhnliche differentialgleichungen mit singularitäten und zugehörigen entwicklungen willkürlicher funktionen. Math. Annal. 68, 220–269 (1910)MathSciNetMATHCrossRefGoogle Scholar
  38. 163.
    Weyl, H.: Über Gewöhnliche Differentialgleichungen mit Singulären Stellen und ihre Eigenfunctionen, pp. 442–467. Göttinger Nachrichten (1910)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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