Abstract
The connection between the integrals of the motion of a quantum system and its Green function is established. The Green function is shown to be the eigenfunction of the integrals of the motion which describe initial points of the system trajectory in the phase space of average coordinates and moments. The explicit expressions for the Green functions of theN-dimensional system with the Hamiltonian which is the most general quadratic form of coordinates and momenta with time-dependent coefficients is obtained in coordinate, momentum, and coherent states representations. The Green functions of the nonstationary singular oscillator and of the stationary Schrödinger equation are also obtained.
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Aronson, E. B., Malkin, I. A., and Man'ko, V. I. (1974).Elementary Particles and Nuclei (in Russian),5, 122.
Bakhrakh, V. L., Khristenko, S. V., and Vetchinkin, S. I. (1972).Teoreticheskaya i Matematicheskaya Fizika,12, 223.
Batalin, I. A., and Fradkin, E. S. (1970).Teoreticheskaya i Matematicheskaya Fizika,5, 190.
Bargmann, V. (1961).Communications on Pure and Applied Mathematics,14, 187.
Barut, A. O., and Girardello, L. (1971).Communications on Mathematical Physics,21, 41.
Berezin, F. A. (1966).The Second Quantization Method, Academic Press, New York.
Boon, M. H., and Seligman, T. H. (1973).Journal of Mathematical Physics,14, 1224.
Chernikov, N. A. (1967).Zhurnal Eksperimentalnoy i Teoreticheskoy Fiziki,53, 1006.
Dimashko, Yu. A., and Granovsky, Ya. I. (1974).Ukrainsky Fizichesky Zhurnal,19, 1456.
Dodonov, V. V., Malkin, I. A., and Man'ko, V. I. (1972).Physics Letters,39A, 377.
Dodonov, V. V., Malkin, I. A., and Man'ko, V. I. (1974a).Physica,72, 597.
Dodonov, V. V., Malkin, I. A., and Man'ko, V. I. (1974b).Nuovo Cimento,24B, 46.
Dodonov, V. V., Malkin, I. A., and Man'ko, V. I. (1974c). Preprint P. N. Lebedev Institute of Physics, No. 106.
Erde'lyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1954).Tables of Integral Transforms, V.I, McGraw-Hill, New York.
Feynman, R. P., and Hibbs, A. R. (1965).Quantum Mechanics and Path Integrals, McGraw-Hill, New York.
Glauber, R. J. (1963).The Physical Review,131, 2766.
Gountaroulis, G. (1972).Physics Letters,40A, 132.
Holz, A. (1970).Lettere al Nuovo Cimento,4, 1319.
Malkin, I. A., and Man'ko, V. I. (1970).Zhurnal Eksperimentalnoy i Teoreticheskoy Fiziki,59, 1746.
Malkin, I. A., and Man'ko, V. I. (1971). Preprint P. N. Lebedev Institute of Physics, No. 15.
Malkin, I. A., Man'ko, V. I., and Trifonov, D. A. (1971).Kratkie Soobsheniya po Fizike, No. 5, pp. 20, 27.
Malkin, I. A., Man'ko, V. I., and Trifonov, D. A. (1973).Journal of Mathematical Physics,14, 576.
Moshinsky, M., and Quesne, C. (1971).Journal of Mathematical Physics,12, 1772.
Popov, M. M. (1973).Problemi Matematicheskoy Fiziki,6, 119.
Wolf, K. B. (1974).Journal of Mathematical Physics,15, 1295.
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Dodonov, V.V., Malkin, I.A. & Man'ko, V.I. Integrals of the motion, green functions, and coherent states of dynamical systems. Int J Theor Phys 14, 37–54 (1975). https://doi.org/10.1007/BF01807990
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DOI: https://doi.org/10.1007/BF01807990