Abstract
The spectral properties of the most general time-dependent potentials of the soliton type described by a self-adjoint operator acting in Hilbert space are discussed. The spectral decomposition for these potentials and the quasispectral decomposition for the Darboux transformation operators are obtained. The coherent states of such systems are examined. Finally, the measure realizing the decomposition of the identity operator in the projectors on the coherent states is calculated.
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References
E. Schrödinger, Naturwissenschaften 14, 664 (1926) [English transl.: Collected Papers on Wave Mechanics, Blackie & Son, London (1928), pp. 41].
A. M. Perelomov, Generalized Coherent States and their Applications, Springer, New York (1986) [Russ. orig. Nauka, Moscow (1987)].
I. A. Malkin and V. I. Man’ko, Dynamic Symmetries and Coherent States of Quantum Systems [in Russian], Nauka, Moscow (1979).
J. R. Klauder and B.-S. Skagerstam, Coherent States; Applications in Physics and Mathematical Physics, World Scientific, Singapore (1985).
V. V. Dodonov and V. I. Man’ko, in Proceedings (Trudy) of the P. N. Lebedev Physics Institute [in Russian], Vol. 183, Moscow (1987), p. 71.
Ya. A. Smorodinskii, A. L. Shelepin, and L. A. Shelepin, Usp. Fiz. Nauk 162(12), 1 (1992) [Sov. Phys. Usp. 35, 1005 (1992)].
V. V. Dodonov, V. I. Man’ko, and O. V. Malkin, in Proceedings (Trudy) of the P. N. Lebedev Physics Institute [in Russian], Vol. 204, Moscow (1988), p. 204.
J. R. Klauder, J. Phys. A 29, L293 (1996).
Ya. I. Granovskii, A. S. Zhedanov, and I. M. Lutsenko, Zh. Éksp. Teor. Fiz. 99, 369 (1991) [Sov. Phys. JETP 72, 205 (1991)].
Ya. I. Granovskii, I. M. Lutzenko, and A. S. Zhedanov, Ann. Phys. (San Diego) 217, 1 (1992).
S. Yu. Dubov, V. M. Eleonskii, and N. E. Kulagin, Zh. Éksp. Teor. Fiz. 102, 814 (1992) [Sov. Phys. JETP 75, 446 (1992)].
V. M. Eleonskii and V. G. Korolev, Zh. Éksp. Teor. Fiz. 110, 1967 (1996) [Sov. Phys. JETP 83, 1084 (1996)].
V. Spiridonov, Phys. Rev. Lett. 69, 398 (1992).
S. Skorik and V. Spiridonov, Lett. Math. Phys. 28, 59 (1993).
A. S. Zhedanov, J. Math. Phys. 34, 2631 (1993).
V. P. Spiridonov, Lett. Math. Phys. 35, 179 (1995).
G. Darboux, Leçons sur la théorie générale des surfaces et les application géométriques du calcul infinitésimale, Deuxiéme partie, Gauthier-Villar et Fils, Paris (1889).
E. L. Ince, Ordinary Differential Equations, Longmans-Green, London (1927).
V. G. Bagrov and B. F. Samsonov, Fiz. Élem. Chastits At. Yadra 28, 951 (1997) [Phys. Part. Nuclei 28, 374 (1997)].
A. B. Shabat, Inverse Probl. 6, 303 (1992).
C. D. J. Fernandez, V. Hussin, and L. M. Nieto, J. Phys. A 27, 3547 (1994).
C. D. J. Fernandez, L. M. Nieto, and O. Rosas-Ortiz, J. Phys. A 28, 2963 (1995).
V. Spiridonov, Phys. Rev. A 52, 1909 (1995).
V. I. Man’ko, G. Marmo, E. C. G. Sudarshan, and F. Zaccaria, Phys. Scr. 55, 528 (1977).
V. G. Bagrov and B. F. Samsonov, Zh. Éksp. Teor. Fiz. 109, 1105 (1996) [JETP 82, 593 (1996)].
V. G. Bagrov and B. F. Samsonov, J. Phys. A 29, 1011 (1996).
V. Matveev and M. Salle, Darboux Transformations and Solitons, Springer, New York (1991).
V. G. Bagrov and B. F. Samsonov, Phys. Lett. A 210, 60 (1996).
P. B. Abraham and H. A. Moses, Phys. Rev. A 22, 1333 (1980).
B. F. Samsonov, J. Math. Phys. 38, 4492 (1997).
B. F. Samsonov, Yad. Fiz. 59, 753 (1996) [Phys. At. Nucl. 59, 720 (1996)].
V. G. Bagrov, B. F. Samsonov, and L. A. Shekoyan, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 1, 84 (1985).
J. Vogel, E. Vogel, and C. Toepffer, Ann. Phys. (San Diego) 164, 463 (1985).
S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Consultants Bureau, New York (1984).
B. A. Dubrovin, T. M. Malanyuk, I. M. Krichever, and V. G. Makhan’kov, Fiz. Élem. Chastits At. Yadra 19, 579 (1988) [Sov. J. Part. Nucl. 19, 252 (1988)].
V. P. Berezovoi and A. I. Pashnev, Teor. Mat. Fiz. 74, 392 (1988).
C. V. Sukumar, J. Phys. A 19, 2297 (1986).
B. F. Samsonov, J. Phys. A 28, 6989 (1995).
W. Miller, Symmetry and Separation of Variables, Addison-Wesley, Reading, MA (1981).
I. M. Gel’fand and G. E. Shilov, Theory of Differential Equations, Academic Press, New York (1967).
F. A. Berezin and M. A. Shubin, The Schrödinger Equation, Kluwer Academic, Boston (1991) [Russ. orig. Moscow Univ. Press, Moscow (1983)].
A. P. Veselov and A. B. Shabat, Funktsional. Anal. Prilozh. 27, 1 (1993).
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis, Academic Press, New York (1978).
I. C. GoKhberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society, Providence, R.I. (1969; reprinted 1978).
I. M. Gel’fand and N. Ya. Vilenkin, Applications of Harmonic Analysis, Academic Press, New York (1964).
F. A. Berezin, Methods of Second Quantization, Academic Press. New York (1963).
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Zh. Éksp. Teor. Fiz. 114, 1930–1943 (December 1998)
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Samsonov, B.F. Coherent states of potentials of soliton origin. J. Exp. Theor. Phys. 87, 1046–1052 (1998). https://doi.org/10.1134/1.558619
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DOI: https://doi.org/10.1134/1.558619