Group Theoretical Methods in Physics pp 245-249 | Cite as

# The rigged Hilbert space and decaying states

Groups and Semigroups in the Description of Decaying Systems

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## Keywords

Energy Operator Resonance Parameter Decay State Complex Eigenvalue Collision Theory
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## Footnotes and References

- 1.J.E. Roberts, Journal Math. Phys. 7, 1097 (1966); A. Böhm, Boulder Lectures in Theoretical Physics, Vol. 9A, 255 (1966).Google Scholar
- 2.A. Böhm, The Rigged HilbertSpace and Quantum Mechanics, Springer Lecture Note 78 (1978); J.E. Roberts, Comm. Math. Phys. 3, 98 (1966);J.P. Antoine, Journal Math. Phys. 10, 53 (1969); 0. Melsheimer, Journal Math. Phys. 15,902 (1974). G. Lassner, Wiss. Z. Karl-Marx-Univ. Leipzig Math.-Naturwiss. R. 22 (1973) H.Z.; A. Böhm, Lectures at the Istanbul Summer Institute 1970 A.0. Barut, editor, Reidel Publishing Co, 1973Google Scholar
- 3.Two distinct suggestions have been made, the first (which constructs a RHS that contains exactly those complex energy eigenvectors which are related to the discrete eigenvalues of H
_{O}= H-V) by H. Baumgärtel in Math. Nachr. 75, 133 (1976) and the second (which defines complex energy eigenvectors that have a BreitWigner energy distribution) by A. Böhm in a manuscript distributed to colleagues in 1976 which will appear as Chapter XXI of reference 4.Google Scholar - 4.A. Böhm, Quantum Mechanics, Springer, New York (1979)Google Scholar
- 5.That.such eigenvectors exist was known to physicists for a long time from their careless manipulation with generators of non-compact subgroups in representations of Lie algebras. The first detaile rigorous discussion of such vectors of SU (1,1) was given by G. Lindblad, B. Nagel, Ann. Inst. Henri Poincaré XIII, 27, 1970. See also G.J. Iverson, Phys. Letters 26B, 229 (1968). Usually one tries to exclude from φ
^{X}all generalised eigenvectors with eigenvalues that do not belong to the spectrum by an appropriate choice of the nuclear topology of ϕ; see reference 12.Google Scholar - 6.a) V. Gorini, G. Parravicini, Proceedings of the VIIth International Group Theory Colloquium. b) E.C.G. Sudarshan, C.B. Chiu, V. Gorini, Phys. Rev. D, to appear. How much the use of the rigged Hilbert space simplifies the description can be seen from a comparison of these two papers. G. Parravicini, V. Gorini, E.C.G. Sudarshan, C.B. Chiu, Journal Math. Phys., to be submitted.Google Scholar
- 7.The association of complex energies with resonances is of course almost as old as quantum mechanics, and other recent suggestions include the use of non-selfadjoint Hamiltonian operators. See e.g. J.S. Howland, J. Math. Anal.Appl. 50, 415 (1975); B. Simon, Annals of Math. 97, (1973), 247; J. Aguilar and J.M. Combes, Comm. Math. Phys. 22, 280 (1977); L.P. Horwitz and I. Sigal, Helv. Phys. Acta, to be published. T.K. Bailey, W.C. Schieve, Nuovo Cimento, Dec. 1978; C. George, F. Henin, F. Mayne, I. Prigogine, Hadron Journal, vol. 1 (1978); A.P. Grecos, I. Prigogine, Proc. Nat. Acad. Sci., USA, 69, 1629 (1972). A. Grossmann, J. Math. Phys. 5, 1025, (1964).Google Scholar
- 8.F. Fonda, G.C. Ghirardi, A. Rimini, Rep. Frog. Phys. 41, 587 (1978), describe the well known deviations from the exponential decay law which however do not apply for φ
^{R}because φ^{R}is not in the domain of the Hilbert space operator H (small times) and φ^{R}was defined in the approximation E_{R}/r » ∞ with the lower bound of the spectrum going to-∞ (large times).Google Scholar - 9.M.L. Goldgerger, K.M. Watson, “Collision Theory,” Wiley, 1964.Google Scholar
- 10.See e.g. H.M. Nussenzveig, “Causality and Dispersion Relations,” Academic Press, 1972, Equation (1.6.6.).Google Scholar
- 11.e.g. H.M. Nussenzveig, “Causality and Dispersion Relations," Academic Press, 1972, Equation (2.8.16).Google Scholar
- 12.K. Napiorkowski, Bulletin of the Polish Academy of Sciences 22, 1215 (1974); 23, 251 (1975).Google Scholar

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© Springer-Verlag 1979