Abstract
The rigged Hilbert space formalism of quantum mechanics provides a framework in which one can identify resonance states and obtain the typical exponential decay law. However, there remain questions of the interpretation and extraction of physical information through the calculation of expectation values of observables. The Lax-Phillips scattering theory provides a mathematical construction in which resonances are assigned with states in a Hilbert space, thus no such difficulties arise. The original Lax-Phillips structure is inapplicable within standard nonrelativistic quantum theory. Through the powerful theory of H p spaces certain relations between the two theories are uncovered, which suggest that a search for a “unifying” framework might prove useful.
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References
Bailey, T. and Schieve, W. C. (1978). Complex energy eigenstates in quantum decay models. Nuovo Cimento A 47, 231.
Baumgartel, W. (1975). Partial resolvent and spectral concentration. Math. Nachr. 69, 107.
Bohm, A. (1979a). The rigged Hilbert space and decaying states. In Group Theoretical Methods in Physics (Proc. Austin 1978), Lecture Notes in Physics, Vol. 94, Springer-Verlag, Berlin, pp. 245-249.
Bohm, A. (1979b). Gamow states vectors as functionals over subspace of the nuclear space. Letters on Mathematical Physics 3, 455.
Bohm, A. (1980). Decaying states in the rigged Hilbert space formulation of quantum mechanics. Journal of Mathematical Physics 21, 1040.
Bohm, A. (1981). Resonance poles and Gamow vectors in the rigged Hilbert space formulation of quantum mechanics. Journal of Mathematical Physics 22, 2813.
Bohm, A. (1986). Quantum Mechanics: Foundations and Applications, Springer, Berlin.
Bohm, A. and Gadella, M. (1989). Dirac Kets, Gamow Vectors and Gel'fand Triplets, Lecture notes in Physics, Vol. 348, Springer-Verlag, Berlin.
Bohm, A., Gadella, M., and Mainland, G. B (1989). Gamow vectors and decaying states. American Journal of Physics 57, 1105.
Cornfield, I. P., Formin, S. V., and Sinai, Y. G. (1982). Ergodic Theory, Springer-Verlag, Berlin.
Duren, P. L. (1970). Theory of H p Spaces, Academic Press, New York.
Eisenberg, E. and Horwitz, L. P. (1997). Time, irreversibility, and unstable systems in quantum physics, Advances in Chemical Physics, 99, 245.
Fourès, Y. and Segal, I. E. (1955). Causality and analyticity. Transactions of the American Mathematical Society 78, 385.
Friedrichs, K. O. (1948). On the perturbation of continuous spectra. Communications in Pure and Applied Mathematics 1, 361.
Gadella, M. (1983a). A rigged Hilbert space of Hardy-class functions: Applications to resonances. Journal of Mathematical Physics 24, 1462.
Gadella, M. (1983b). A description of virtual scattering states in the rigged Hilbert space formulation of quantum mechanics. Journal of Mathematical Physics 24, 2142.
Gadella, M. (1984). On the RHS description of resonances and virtual states. Journal of Mathematical Physics 25, 2481.
Gel'fand, I. M. and Vilenkin, Ya. N. (1964). Generalized Functions IV, Academic Press, New York.
Hoffman, K. (1962). Banach Spaces of Analytic Functions, Prentice-Hall, Englewood Cliffs, NJ.
Horwitz, L. P. and Piron, C. (1993). The unstable system and irreversible motion in quantum theory. Helvetica Physica Acta 66, 693.
Horwitz, L. P. and Sigal, I. M. (1978). On a mathematical model for nonstationary physical systems. Helvetica Physica Acta 51, 685.
Lax, P. D. and Phillips, R. S. (1967). Scattering Theory, Academic Press, New York.
Nikol'skii, N. K. (1986). Treatise on the Shift Operator, Springer-Verlag, Berlin.
Paley, R. E. A. C. and Wiener, N. (1934). Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publications, Vol. 19, New York.
Parravicini, G., Gorini, V., and Sudarshan, E. C. G (1980). Resonances, scattering theory, and rigged Hilbert spaces. Journal of Mathematical Physics 21, 2208.
Rosenblum, M. and Rovnyak, J. (1985). Hardy Classes and Operator Theory, Oxford University Press, New York; and references therein.
Strauss, Y. and Horwitz, L. P., and Eisenberg, E. (2000). Representation of quantum mechanical resonances in Lax-Phillips Hilbert space. Journal of Mathematical Physics 41, 8050.
Strauss, Y. and Horwitz, L. P. (2000). Representaton of the resonance of a relativistic quantum field theoretical Lee-Friedrichs model in Lax-Phillips scattering theory. Foundations of Physics 30, 653.
Sz.-Nagy, B. and Foias, C. (1970). Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam.
Van Winter, C. (1971). Fredholm equations on a Hilbert space of analytic functions. Transactions of the American Mathematical Society 162, 103.
Weisskopf, V. F. and Wigner, E. P. (1930). Berechung der naturlichen Linienbreite auf grund der diracschen lichttheori. Zeitschrift fur Physik 63, 54.
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Strauss, Y. Resonances in the Rigged Hilbert Space and Lax-Phillips Scattering Theory. International Journal of Theoretical Physics 42, 2285–2315 (2003). https://doi.org/10.1023/B:IJTP.0000005959.97056.8b
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DOI: https://doi.org/10.1023/B:IJTP.0000005959.97056.8b