Exponential Decay in Quantum Mechanics

  • V. Kruglov
  • K. A. Makarov
  • B. Pavlov
  • A. Yafyasov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7160)


First of all: disregard all of so called “preliminary physical expectations”, because all these expectations are just a pseudonym for prejudices elaborated by the older generation. These words belong to Dirac, not to me. The right way of creating new physics is different: one should begin with a beautiful mathematical idea. But it should be really beautiful! No special relations to physics is compulsory. But if it is really beautiful, it will certainly match useful physical applications, though it is not predefined, what sort of applications and where: it depends on physical consequenceswhich may be extracted from the mathematical scheme.


Invariant Subspace Cooper Pair Blaschke Product Cauchy Data Hardy Class 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dirac, P.A.M.: Development of the physicists’s conception of Nature. In: Mehra, J. (ed.) The Physicists Conception of Nature, pp. 1–14. D. Reidel Publ. (1973)Google Scholar
  2. 2.
    von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton (1996) (12th printing)zbMATHGoogle Scholar
  3. 3.
    Gamow, G.: Zur Quantentheorie des Atomkernes. Zeitshrift für Physik 51, 204–2012 (1928)CrossRefzbMATHGoogle Scholar
  4. 4.
    Weisskopf, V.E., Wigner, E.P.: Zeitshrift für Physik 63, 54 (1930); 65, 18 (1930)Google Scholar
  5. 5.
    Fock, V.A., Krylov, V.A.: Journal of Experimental and Theoretical Physics (USSR) 17, 93 (1947)Google Scholar
  6. 6.
    Fock, V.A.: Selected Works. Quantum Mechanics and Quantum Field Theory. In: Faddeev, L.D., Khalfin, L.A., Komarov, I.V. (eds.), Chapman & Hall/CRC (2004)Google Scholar
  7. 7.
    Khalfin, L.: On the theory of decay of a quasi-stationary state. Soviet Phys. Doklady 2, 340 (1958)zbMATHGoogle Scholar
  8. 8.
    Sakurai, J.: Modern Quantum Mechanics. Revised Edition. Addison-Wesley (1994)Google Scholar
  9. 9.
    Titchmarsh, E.C.: Eigenfunction Expansion Associated with Second-order Differential Equations, Part 1. Clarendon Press, Oxford (1962)zbMATHGoogle Scholar
  10. 10.
    Lax, P., Phillips, R.: Scattering Theory. Academic Press, New York (1967)zbMATHGoogle Scholar
  11. 11.
    Hegerfeldt, G.C.: Causality, particle localization and positivity of the energy. In: Irreversibility and Causality: Semigroups and Rigged Hilbert Spaces. Lecture Notes in Physics, vol. 504, pp. 238–245 (1998)Google Scholar
  12. 12.
    Beurling, A.: On two problems concerning linear transformations in Hilbert Space. Acta. Math. 81, 239–255 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nagy, B.S., Foiaş, C.: Harmonic Analysis of Operators on Hilbert Space. Akademiai Kiado, Budapest (1970)zbMATHGoogle Scholar
  14. 14.
    Koosis, P.: Introduction to Hp Spaces, 2nd edn. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  15. 15.
    Pavlov, B.: Spectral analysis of a dissipative singular Schrödinger operator in terms of a functional model. In: Shubin, M. (ed.) Partial Differential Equations. Encyclopedia of Mathematical Sciences, vol. 65, pp. 87–153. Springer, Heidelberg (1995)Google Scholar
  16. 16.
    Nikol’skii, N.K., Khruschev, S.V.: A functional model and some problems of the spectral theory of functions. Trudy Mat. Inst. Steklov. 176, 97–210, 327 (1987)MathSciNetGoogle Scholar
  17. 17.
    Krein, M.G.: Selected Works. II: Banach Spaces and Operator Theory, Natsional’naya Akademiya Nauk Ukrainy, Institut Matematiki, Kiev (1996) (Russian)Google Scholar
  18. 18.
    Krein, M.G.: Selected Works. III. Topics in Differential and Integral Equations and Operator Theory. In: Gohberg, I. (ed.), Birkhauser Verlag, Basel (1983)Google Scholar
  19. 19.
    Adamjan, V.M., Arov, D.Z.: On scattering operators and contraction semigroups in Hilbert space. Dokl. Akad. Nauk SSSR 165, 9–12 (1965) (Russian)MathSciNetGoogle Scholar
  20. 20.
    Livshits, M.S.: Method of non-selfadjoint operators in the theory of waveguides. Radio Engineering and Electronic Physics, American Institute of Electrical Engineers 1, 260–275 (1962)Google Scholar
  21. 21.
    Pavlov, B.: The theory of extensions and explicitly-soluble models. Russian Math. Surveys 42(6), 127–168 (1987)CrossRefzbMATHGoogle Scholar
  22. 22.
    Flesia, C., Piron, C.: Helv. Phys. Acta 57, 697 (1984)Google Scholar
  23. 23.
    Horwitz, L.P., Piron, C.: Helv. Phys. Acta 66, 694 (1993)Google Scholar
  24. 24.
    Strauss, Y., Horwitz, L.P., Eisenberg, E.: Representation of quantum mechanical resonances in the lax-Phillips Hilbert space. Journal of Mathematical Physics 41, 12 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Baumgartel, H.: Gamov vectors for resonances: a Lax-Phillips point of view. International Journal of Theoretical Physics 46(8), 1960–1985 (2007)CrossRefzbMATHGoogle Scholar
  26. 26.
    Baumgartel, H.: Resonances of quantum-mechanical scattering systems and lax-Phillips scattering theory. Journal of Mathematical Physics 51(113508), 1–20 (2010)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ginzburg, V., Landau, L.: Toward the superconductivity theory. Zhurnal Eksp. Yheoret. Physics 29, 1064 (1950) (Russian)Google Scholar
  28. 28.
    Okun, L.B.: Leptons and Quarks. North Holland, Amsterdam (1981)Google Scholar
  29. 29.
    Ponomarev, A., Yudovich, M., Gruzdev, M., Yudovich, V.: Theoretical estimations of topological factor in interaction of the nano-particles with electromagnetic waves. Scientific Israel-Technological Advancements 11(3), 20–26 (2009)Google Scholar
  30. 30.
    Gribov, V.N.: Quantum Electrodynamics, Moscow, Igevsk (2001) (Russian)Google Scholar
  31. 31.
    Misra, B., Sudarshan, E.C.G.: The Zeno paradox in quantum theory. Journal of Mathematical Physics 18(4), 753–756 (1977)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Davydov, A.S.: Quantum Mechanics, ch. IX, Section 80. Pergamon (1965),Google Scholar
  33. 33.
    Krasnosel’skij, M.A.: On self-adjoint extensions of Hermitian operators. Ukrainskij Mat. Journal 1, 21 (1949) (Russian)MathSciNetGoogle Scholar
  34. 34.
    Shirokov, J.: Strongly singular potentials in three-dimensional quantum mechanics. Teor. Mat. Fiz. 42(1), 45–49 (1980) (Russian)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Albeverio, S., Kurasov, P.: Singular Perturbations of Differential Operators. London Math. Society Lecture Note Series, vol. 271. Cambridge University Press (2000)Google Scholar
  36. 36.
    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space, vol. 1. Frederick Ungar, Publ., New-York (1966)zbMATHGoogle Scholar
  37. 37.
    Adamyan, V.A., Calude, C.S., Pavlov, B.S.: Transcending the limits of Turing computability. In: Hida, T., Saitô, K., Si, S. (eds.) Quantum Information Complexity. Proceedings of Meijo Winter School 2003, pp. 119–137. World Scientific, Singapore (2004)Google Scholar
  38. 38.
    Pavlov, B.: A star-graph model via operator extension. Mathematical Proceedings of the Cambridge Philosophical Society 142(02), 365–384 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Yafyasov, A., Martin, G., Pavlov, B.: Resonance one-body scattering on a junction. Nanosystems: Physics, Chemistry, Mathematics 1(1), 108–147 (2010)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • V. Kruglov
    • 1
  • K. A. Makarov
    • 2
  • B. Pavlov
    • 3
    • 4
  • A. Yafyasov
    • 4
  1. 1.Department of PhysicsUniversity of AucklandNew Zealand
  2. 2.Department of MathematicsUniversity of MissouriColumbiaUSA
  3. 3.The New Zealand Institute for Advanced StudyMassey UniversityAucklandNew Zealand
  4. 4.V.A. Fock Institute for Physics, Department of PhysicsSt. Petersburg UniversityRussia

Personalised recommendations