On the number of negative eigenvalues of a~partial integral operator
Matematičeskie trudy, Tome 17 (2014) no. 1, pp. 128-144.

Voir la notice de l'article provenant de la source Math-Net.Ru

We find the lower boundary for the essential spectrum of a Fredholm type partial integral operator $H$. We also obtain an estimate for the number of eigenvalues below this boundary.
@article{MT_2014_17_1_a5,
     author = {R. R. Kucharov and Yu. Kh. Eshkabilov},
     title = {On the number of negative eigenvalues of a~partial integral operator},
     journal = {Matemati\v{c}eskie trudy},
     pages = {128--144},
     publisher = {mathdoc},
     volume = {17},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MT_2014_17_1_a5/}
}
TY  - JOUR
AU  - R. R. Kucharov
AU  - Yu. Kh. Eshkabilov
TI  - On the number of negative eigenvalues of a~partial integral operator
JO  - Matematičeskie trudy
PY  - 2014
SP  - 128
EP  - 144
VL  - 17
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MT_2014_17_1_a5/
LA  - ru
ID  - MT_2014_17_1_a5
ER  - 
%0 Journal Article
%A R. R. Kucharov
%A Yu. Kh. Eshkabilov
%T On the number of negative eigenvalues of a~partial integral operator
%J Matematičeskie trudy
%D 2014
%P 128-144
%V 17
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MT_2014_17_1_a5/
%G ru
%F MT_2014_17_1_a5
R. R. Kucharov; Yu. Kh. Eshkabilov. On the number of negative eigenvalues of a~partial integral operator. Matematičeskie trudy, Tome 17 (2014) no. 1, pp. 128-144. http://geodesic.mathdoc.fr/item/MT_2014_17_1_a5/

[1] Aleksandrov V. M., Kovalenko E. V., “Ob odnom klasse integralnykh uravnenii smeshannykh zadach mekhaniki sploshnykh sred”, Dokl. AN SSSR, 252:2 (1980), 324–328 | MR | Zbl

[2] Aleksandrov V. M., Kovalenko E. V., “O kontaktnom vzaimodeistvii tel s pokrytiyami”, Dokl. AN SSSR, 275:4 (1984), 827–830 | MR | Zbl

[3] Akhiezer A. I., Baryakhtar V. G., Peletminskii S. V., Spinovye volny, Nauka, M., 1967

[4] Vekua I. N., Novye metody resheniya ellipticheskikh uravnenii, OGIZ, M.–L., 1948 | MR

[5] Gursa E., Kurs matematicheskogo analiza, v. 3, Ch. 2, Integralnye uravneniya. Variatsionnoe ischislenie, GTTI, M.–L., 1934

[6] Zhukov Yu. V., “Teorema Iorio–O'Kerrola dlya $N$-chastichnogo reshetchatogo gamiltoniana”, TMF, 107:1 (1996), 75–85 | DOI | MR | Zbl

[7] Kalitvin A. S., “O nekotorykh klassakh chastichno integralnykh uravnenii v aerodinamike”, Sostoyanie i perspektiva razvitiya nauki i tekhniki, LGPU, Lipetsk, 1994, 210–212

[8] Kalitvin A. S., Lineinye operatory s chastnymi integralami, TsChKI, Voronezh, 2000

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

[10] Lakshtanov E. A., Minlos R. A., “Spektr dvukhchastichnykh svyazannykh sostoyanii transfer-matrits gibbsovskikh polei (uedinennoe svyazannoe sostoyanie)”, Funkts. analiz i ego pril., 38:3 (2004), 52–69 | DOI | MR | Zbl

[11] Malyshev V. A., Minlos R. A., “Klasternye operatory”, Tr. seminara im. I. G. Petrovskogo, 9, Izd-vo Mosk. un-ta, M., 1983, 63–80 | MR

[12] Manzhirov A. V., “Ob odnom metode resheniya dvumernykh integralnykh uravnenii osesimmetrichnykh kontaktnykh zadach dlya tel so slozhnoi reologiei”, Prikl. mat. i mekh., 49:6 (1985), 1019–1025 | MR | Zbl

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

[14] Myuntts G., Integralnye uravneniya, v. 1, Lineinye uravneniya, Gostekhteorizdat, L.–M., 1934

[15] Rasulov T. Kh., “Asimptotika diskretnogo spektra odnogo modelnogo operatora, assotsiirovannogo s sistemoi trekh chastits na reshetke”, TMF, 163:1 (2010), 34–44 | DOI | MR | Zbl

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

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

[18] Trikomi F., Integralnye uravneniya, Izd-vo inostr. lit., M., 1960 | MR

[19] Faddeev L. D., “O modeli Fridrikhsa v teorii vozmuschenii nepreryvnogo spektra”, Kraevye zadachi matematicheskoi fiziki. 2, Sb. 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

[20] Eshkabilov Yu. Kh., “O spektre tenzornoi summy kompaktnykh operatorov”, Uzbekskii mat. zhurn., 2005, no. 3, 104–112 | MR

[21] Eshkabilov Yu. Kh., “Vozmuschenie spektra operatora umnozheniya na funktsiyu s chastnym integralnym operatorom”, Vestnik nats. un-ta Uzbekistana, 2006, no. 2, 17–21 | MR

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

[23] Eshkabilov Yu. Kh., “Suschestvennyi i diskretnyi spektry chastichno integralnykh operatorov”, Matem. tr., 11:2 (2008), 187–203 | MR

[24] Eshkabilov Yu. Kh., “Effekt Efimova dlya odnogo modelnogo “trekhchastichnogo” diskretnogo operatora Shredingera”, TMF, 164:1 (2010), 78–87 | DOI | Zbl

[25] Eshkabilov Yu. Kh., “O beskonechnosti diskretnogo spektra operatorov v modeli Fridrikhsa”, Matem. tr., 14:1 (2011), 195–211 | MR

[26] Eshkabilov Yu. Kh., Kucharov R. R., “O suschestvennom i diskretnom spektrakh trekhchastichnogo operatora Shredingera na reshetke”, TMF, 170:3 (2012), 409–422 | DOI | Zbl

[27] Albeverio S., Lakaev S. N., Muminov Z. I., “On the number of eigenvalues of a model operator associated to a system of three-particles on lattices”, Russ. J. Math. Phys., 14:4 (2007), 377–387 | DOI | MR | Zbl

[28] Appell J., Kalitvin A. S., Nashed M. Z., “On some partial integral equations arising in the mechanics of solids”, Z. Angew. Math. Mech., 79:10 (1999), 703–713 | 3.0.CO;2-W class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[29] Appell J., Kalitvin A. S., Zabrejko P. P., Partial Integral Operators and Integro-Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics, 230, Marcel Dekker, Inc., New York, 2000 | MR | Zbl

[30] Kalitvin A. S., Zabrejko P. P., “On the theory of partial integral operators”, J. Integral Equations Appl., 3:3 (1991), 351–382 | DOI | MR | Zbl

[31] Malyshev V. A., Minlos R. A., “Invariant spaces of clustering operator. I”, J. Statist. Phys., 21:3 (1979), 231–242 | DOI | MR

[32] Malyshev V. A., Minlos R. A., “Invariant spaces of clustering operator. II”, Comm. Math. Phys., 82:2 (1981/82), 211–226 | DOI | MR | Zbl