Abstract
We consider matrix structures in the quantum N-body problem that generalize the Faddeev components for resolvents, T-matrices, and eigenfunctions of the continuous spectrum. We write matrix equations for the introduced components of T-matrices and resolvents and use these equations to obtain matrix operators generalizing the matrix three-particle Faddeev operators to the case of arbitrarily many particles. We determine the eigenfunctions of the continuous spectrum of these matrix operators.
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References
L. D. Faddeev, Mathematical Questions in the Quantum Theory of Scattering for a System of Three Particles [in Russian] (Trudy Mat. Inst. Steklov., Vol. 69), Acad. Sci. USSR, Moscow (1963).
L. D. Faddeev and S. P. Merkuriev, Quantum Scattering Theory for Several Particles Systems [in Russian], Nauka, Moscow (1985); English transl. (Math. Phys. Appl. Math., Vol. 11), Kluwer, Dordrecht (1993).
S. Weinberg, Phys. Rev. B, 133, 232–256 (1964).
L. Rosenberg, Phys. Rev. B, 140, 217–226 (1965).
A. N. Mitra, J. Gillespie, B. Sugar, and N. Panchapakesan, Phys. Rev. B, 140, 1336–1338 (1965).
V. A. Alessandrini, J. Math. Phys., 7, 213–220 (1966).
J. Weyers, Phys. Rev., 145, 1236–1242 (1966).
N. Mishima and J. Takahashi, Prog. Theoret. Phys., 35, 440–451 (1966).
R. Omnes, Phys. Rev., 165, 1265–1272 (1968).
O. A. Yakubovskii, Soviet J. Nucl. Phys., 5, 937–942 (1967).
O. A. Yakubovskii, Trudy Mat. Inst. Steklov., 110, 146–177 (1970).
K. Hepp, Helv. Phys. Acta, 42, 425–458 (1969).
S. P. Merkur’ev and S. L. Yakovlev, Dokl. Akad. Nauk SSSR, 262, 591–594 (1982).
S. P. Merkur’ev and S. L. Yakovlev, Theor. Math. Phys., 56, 673–682 (1983).
S. P. Merkur’ev and S. L. Yakovlev, Soviet J. Nucl. Phys., 39, 1002–1006 (1985).
S. P. Merkuriev, S. L. Yakovlev, and C. Gignoux, Nucl. Phys. A, 431, 125–138 (1984).
A. A. Kvitsinsky, Yu. A. Kuperin, S. P. Merkuriev, A. K. Motovilov, and S. L. Yakovlev, PEPAN, 17, 267–317 (1986).
S. L. Yakovlev, Theor. Math. Phys., 82, 157–169 (1990).
S. P. Merkur’ev, A. K. Motovilov, and S. D. Yakovlev, Theor. Math. Phys., 94, 306–314 (1993).
V. A. Roudnev and S. L. Yakovlev, Soviet J. Nucl. Phys., 58, 1762–1771 (1995).
S. L. Yakovlev, Theor. Math. Phys., 102, 235–244 (1995).
S. L. Yakovlev, Theor. Math. Phys., 107, 835–847 (1996).
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin (1995).
S. L. Yakovlev and Z. Papp, Theor. Math. Phys., 163, 666–676 (2010).
I. M. Narodetskii and O. A. Yakubovskii, “Integral equations of scattering theory for N particles [in Russian],” in: Many-Body Problem in Nuclear Physics, Joint Inst. Nucl. Res., Dubna (1980), pp. 183–226.
M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 3, Scattering Theory, Acad. Press, New York (1979).
H. P. Noyes and H. Fiedeldey, “Calculations of three-nucleon low-energy parameters,” in: Three-Particle Scattering in Quantum Mechanics (J. Gillespie and J. Nuttall, eds.), Benjamin, New York (1968), pp. 195–293.
S. P. Merkuriev, C. Gignoux, and A. Laverne, Ann. Phys., 99, 30–71 (1976).
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Dedicated to Academician L. D. Faddeev on the occasion of his 80th birthday
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 181, No. 1, pp. 218–240, October, 2014.
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Yakovlev, S.L. Quantum N-body problem: Matrix structures and equations. Theor Math Phys 181, 1317–1338 (2014). https://doi.org/10.1007/s11232-014-0215-5
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DOI: https://doi.org/10.1007/s11232-014-0215-5