Communications in Mathematical Physics

, Volume 173, Issue 3, pp 647–673

On the global existence of Bohmian mechanics

  • K. Berndl
  • D. Dürr
  • S. Goldstein
  • G. Peruzzi
  • N. Zanghì


We show that the particle motion in Bohmian mechanics, given by the solution of an ordinary differential equation, exists globally: For a large class of potentials the singularities of the velocity field and infinity will not be reached in finite time for typical initial values. A substantial part of the analysis is based on the probabilistic significance of the quantum flux. We elucidate the connection between the conditions necessary for global existence and the self-adjointness of the Schrödinger Hamiltonian.


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  1. 1.
    Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable models in quantum mechanics. Berlin, Heidelberg, New York: Springer, 1988Google Scholar
  2. 2.
    Arnold, V.I.: Small denominators and problems of stability of motion in classical and celestial mechanics. Russ. Math. Surv.18(6), 85–191 (1963)Google Scholar
  3. 3.
    Arnold, V.I.: Mathematical methods in classical mechanics. Berlin, Heidelberg, New York: Springer, 1989Google Scholar
  4. 4.
    Bell, J.S.: Speakable and unspeakable in quantum mechanics. Cambridge: Cambridge University Press, 1987Google Scholar
  5. 5.
    Beraha, L.: Diplom thesis. Universität München, 1994Google Scholar
  6. 6.
    Bochner, S., von Neumann, J.: On compact solutions of operational-differential equations I. Ann. Math.36, 255–291 (1935)Google Scholar
  7. 7.
    Bohm, D.: A suggested interpretation of the quantum theory in terms of “hidden” variables, Parts I and II. Phys. Rev.85, 166–179 and 180–193 (1952)Google Scholar
  8. 8.
    Bohm, D., Hiley, B.J.: The undivided universe: An ontological interpretation of quantum theory. London: Routledge, 1993Google Scholar
  9. 9.
    Carlen, E.A.: Conservative diffusions. Commun. Math. Phys.94, 293–315 (1984)Google Scholar
  10. 10.
    Case, K.M.: Singular potentials. Phys. Rev.80, 797–806 (1950)Google Scholar
  11. 11.
    Daumer, M., Dürr, D., Goldstein, S., Zanghì, N.: On the role of operators in quantum theory. In preparationGoogle Scholar
  12. 12.
    Diacu, F.N.: Singularities of theN-body problem. Montréal: Les Publications CRM, 1992Google Scholar
  13. 13.
    Dürr, D., Goldstein, S., Zanghì, N.: Quantum equilibrium and the origin of absolute uncertainty. J. Stat. Phys.67, 843–907 (1992)Google Scholar
  14. 14.
    Dürr, D., Goldstein, S., Zanghì, N.: Quantum mechanics, randomness, and deterministic reality. Phys. Lett. A172, 6–12 (1992)Google Scholar
  15. 15.
    Faris, W.G.: Self-adjoint operators. Lecture Notes in Mathematics433. Berlin: Springer, 1975Google Scholar
  16. 16.
    Gerver, J.L.: The existence of pseudocollisions in the plane. J. Differential Equations89, 1–68 (1991)Google Scholar
  17. 17.
    Holland, P.R.: The quantum theory of motion. Cambridge: Cambridge University Press, 1993Google Scholar
  18. 18.
    Hunziker, W.: On the space-time behavior of Schrödinger wavefunctions. J. Math. Phys.7, 300–304 (1966)Google Scholar
  19. 19.
    Jörgens, K., Rellich, F.: Eigenwerttheorie gewöhnlicher Differentialgleichungen. Berlin, Heidelberg, New York: Springer, 1976Google Scholar
  20. 20.
    Kato, T.: Fundamental properties of Hamiltonian operators of Schrödinger type. Trans. Am. Math. Soc.70, 195–211 (1951)Google Scholar
  21. 21.
    Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer, 1966Google Scholar
  22. 22.
    Kemble, E.C.: The fundamental principles of quantum mechanics. New York: McGrawHill, 1937Google Scholar
  23. 23.
    Landau, L.D., Lifshitz, E.M.: Quantum mechanics, nonrelativistic theory. Oxford: Pergamon Press, 1958Google Scholar
  24. 24.
    Mather, J., McGehee, R.: Solutions of the collinear four body problem which become unbounded in finite time. In: Moser, J. (ed.): Dynamical systems: Theory and applications. Berlin, Heidelberg, New York: Springer, 1975Google Scholar
  25. 25.
    Mather, J.: Private communicationGoogle Scholar
  26. 26.
    Moser, J.: Stable and random motions in dynamical systems. Princeton: Princeton University Press, 1973Google Scholar
  27. 27.
    Nelson, E.: Feynman integrals and Schrödinger equation. J Math. Phys.5, 332–343 (1964)Google Scholar
  28. 28.
    Nelson, E.: Quantum fluctuations. Princeton: Princeton University Press, 1985Google Scholar
  29. 29.
    Radin, C., Simon, B.: Invariant domains for the time-dependent Schrödinger equation. J. Differential Equations29, 289–296 (1978)Google Scholar
  30. 30.
    Rauch, J.: Partial Differential Equations. Berlin, Heidelberg, New York: Springer, 1991Google Scholar
  31. 31.
    Reed, M., Simon, B.: Methods of modern mathematical physics I. Orlando: Academic Press, 1980Google Scholar
  32. 32.
    Reed, M., Simon, B.: Methods of modern mathematical physics II. San Diego: Academic Press, 1975Google Scholar
  33. 33.
    Rudin, W.: Functional analysis. New York: McGraw Hill, 1991Google Scholar
  34. 34.
    Saari, D.G.: A global existence theorem for the four-body problem of Newtonian mechanics. J. Differential Equations26, 80–111 (1977)Google Scholar
  35. 35.
    Seba, P.: Schrödinger particle on a half line. Lett. Math. Phys.10, 21–27 (1985)Google Scholar
  36. 36.
    Simon, B.: Quantum mechanics for Hamiltonians defined as quadratic forms. Princeton: Princeton University Press, 1971Google Scholar
  37. 37.
    Weidmann, J.: Linear operators in Hilbert spaces. Berlin, Heidelberg, New York: Springer, 1980Google Scholar
  38. 38.
    Xia, Z.: The existence of noncollision singularities in Newtonian systems. Ann. Math.135, 411–468 (1992)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • K. Berndl
    • 1
  • D. Dürr
    • 1
  • S. Goldstein
    • 2
  • G. Peruzzi
    • 3
  • N. Zanghì
    • 4
  1. 1.Mathematisches Institut der Universität MünchenMünchenGermany
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA
  3. 3.Sezione INFN FirenzeFirenzeItaly
  4. 4.Dipartimento di FisicaUniversità di Genova, Sezione INFN GenovaGenovaItaly

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