On infinity of the discrete spectrum of operators in the Friedrichs model

Yu. Kh. Eshkabilov

Matematičeskie trudy, Tome 14 (2011) no. 1, pp. 195-211.

Résumé

The discrete spectrumof selfadjoint operators in the Friedrichs model is studied. Necessary and sufficient conditions of existence of infinitely many eigenvalues in the Friedrichs model are presented. A discrete spectrum of a model three-particle discrete Schrödinger operator is described.

Bibliographie

[1] Berezin F. A., Shubin M. A., Uravnenie Shredingera, MGU, M., 1983 | MR

[2] Efimov V. N., “Svyazannye sostoyaniya trekh rezonansov vzaimodeistvuyuschikh chastits”, Yadernaya fizika, 12:5 (1970), 1080–1091

[3] Imomkulov S. A., Lakaev S. N., “Diskretnyi spektr odnomernoi modeli Fridrikhsa”, Dokl. AN UzSSR, 1988, no. 7, 9–11 | MR | Zbl

[4] Ladyzhenskaya O. A., Faddeev L. D., “K teorii vozmuschenii nepreryvnogo spektra”, Dokl. AN SSSR, 145:2 (1962), 301–304

[5] Lakaev S. N., “O diskretnom spektre obobschennoi modeli Fridrikhsa”, Dokl. AN UzSSR, 1979, no. 4, 9–10 | MR | Zbl

[6] Lakaev S. N., “Nekotorye spektralnye svoistva obobschennoi modeli Fridrikhsa”, Tr. seminara I. G. Petrovskogo, 11, 1986, 210–238 | Zbl

[7] Lakaev S. N., “O beskonechnom chisle trekhchastichnykh svyazannykh sostoyanii sistemy trekh kvantovykh reshetchatykh chastits”, TMF, 89:1 (1991), 94–104 | MR

[8] Lakaev S. N., “Ob effekte Efimova v sisteme trekh odinakovykh kvantovykh chastits”, Funkts. analiz i ego pril., 27:3 (1993), 15–28 | MR | Zbl

[9] Lakaev S. N., Minlos R. A., “O svyazannykh sostoyaniyakh klasternogo operatora”, TMF, 39:1 (1979), 83–93 | MR

[10] Merkurev S. P., Faddeev L. D., Kvantovaya teoriya rasseyaniya dlya sistem neskolkikh chastits, Nauka, M., 1985 | MR

[11] Minlos R. A., Sinai Ya. G., “Issledovanie spektrov stokhasticheskikh operatorov, voznikayuschikh v reshetchatykh modelyakh gaza”, TMF, 2:2 (1970), 230–243 | MR

[12] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, v. 1, Funktsionalnyi analiz, Mir, M., 1977 | MR

[13] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, v. 4, Analiz operatorov, Mir, M., 1982 | MR

[14] Faddeev L. D., “O modeli Fridrikhsa v teorii vozmuschenii nepreryvnogo spektra”, Kraevye zadachi matematicheskoi fiziki. 2, Sbornik rabot. Posvyaschaetsya pamyati V. A. Steklova v svyazi so stoletiem so dnya ego rozhdeniya, Tr. MIAN SSSR, 73, Nauka, M.–L., 1964, 292–313 | MR | Zbl

[15] Eshkabilov Yu. Kh., “Ob odnom diskretnom “trekhchastichnom” operatore Shredingera v modeli Khabbarda”, TMF, 149:2 (2006), 228–243 | MR

[16] Yafaev D. R., “K teorii diskretnogo spektra trekhchastichnogo operatora Shredingera”, Matem. sb., 94(136):4(8) (1974), 567–593 | MR | Zbl

[17] Abdullaev J. I., Lakaev S. N., “On the spectral properties of the matrix-valued Friedrichs model”, Many-Particle Hamiltonians: Spectra and Scattering, Adv. Soviet Math., 5, Amer. Math. Soc., Providence, RI, 1991, 1–37 | MR

[18] Amado R. D., Noble J. V., “Efimov's effect: a new pathology of three-particle systems. I”, Phys. Lett. B, 35:1 (1971), 25–27 | DOI | MR

[19] Amado R. D., Noble J. V., “Efimov's effect: a new pathology of three-particle systems. II”, Phys. Rev. D, 5:8 (1972), 1992–2002 | DOI

[20] Friedrichs K. O., “Über die Spectralzerlegung eines Integral Operators”, Math. Ann., 115:1 (1938), 249–272 | DOI | MR | Zbl

[21] Mattis D. C., “The few-body problem on a lattice”, Rev. Modern Phys., 58:2 (1986), 361–379 | DOI | MR | Zbl

[22] Mogilner A. I., “The problem of a few quasi-particles in SSPh”, Application of Self-Adjoint Extensions in Quantum Physics, Lecture Notes in Physics, 324, eds. P. Exner, P. Seba, Springer-Verlag, Berlin, 1988

[23] Ovchinnikov Yu. N., Sigal I. M., “Number of bound states of three-body systems and Efimov's effect”, Ann. Physics, 123:2 (1979), 274–295 | DOI | MR

[24] Tamura H., “The Efimov effect of three-body Schrödinger operators”, J. Funct. Anal., 95:2 (1991), 433–459 | DOI | MR | Zbl