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On the finiteness of discrete spectrum in the n-particle problem. (English) Zbl 0581.46063
Authors’ summary: ”In spaces of functions of any given symmetry, sufficient conditions have been obtained for the discrete spectrum finiteness of the energy operators of many particle quantum systems in the absence of external fields.
The most general case is considered when the beginning of the essential spectrum of the Hamiltonian is defind by breaking the initial system into any number of stable subsystems. Results obtained are applicable in particular to systems with a short-range and to systems with long-range action (in the case of interaction compensation).”
Reviewer: P.Hillion

##### MSC:
 46N99 Miscellaneous applications of functional analysis 47A70 (Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces 47A40 Scattering theory of linear operators 81U10 $$n$$-body potential quantum scattering theory
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##### References:
 [1] Hunziker, W., Acta physica austriaca, 17, 43, (1976), Supplementum [2] Iorgens, K.; Weidmann, I., Spectral properties of Hamiltonian operators, () [3] Antonets, M.A.; Zhislin, G.M.; Shereshevskij, I.A., Appendix to the Russian translation of ref. 2, (1976), Mir Moscow [4] Simon, B., Comm. math. phys., 55, 259, (1977) [5] Enss, W., Comm. math. phys., 52, 233, (1977) [6] Zhislin, G.M., Izvestiya AN SSSR (ser. math), 33, 590, (1969) [7] Yafaev, D.R., Izvestiya AN SSSR (ser. math), 40, 908, (1976) [8] Vugal’ter, S.A.; Zhislin, G.M., Theor. math. fiz., 32, 70, (1977) [9] Efimov, V.N., Yadernaya fizika, 12, 1880, (1970) [10] Yafaev, D.R., Math. sbornik, 94, 567, (1974) [11] Yafaev, D.R., Theor. math. fiz., 25, 185, (1975) [12] Combescure, M.; Gimbre, L., Ann. of phys., 101, 355, (1976) [13] Vugal’ter, S.A., Funk. analys., 12, 74, (1978) [14] Iorio, R.I., Comm. mat. phys., 62, 201, (1978) [15] Sigal, I.M., Comm. mat. phys., 48, 137, (1976) [16] Sigal, I.M., Comm. mat. phys., 48, 155, (1976) [17] Aguilar, I.; Combes, I.M., Comm. mat. phys., 22, 269, (1971) [18] Sigalov, A.G.; Sigal, I.M., Theor. mat. phys., 5, 73, (1970) [19] Zhislin, G.M., Theoret. mat. phys., 21, 60, (1974) [20] Birman, M.S., Mat. sbornik, 55, 125, (1961) [21] Zhislin, G.M., Uspekhi mat. nauk, 19, 155, (1964) [22] Courant, R.; Hilbert, D., Methods of mathematical physics, (1962), Wiley (Interscience) New York · Zbl 0729.00007
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