Essential and discrete spectra of partially integral operators

Yu. Kh. Eshkabilov

Matematičeskie trudy, Tome 11 (2008) no. 2, pp. 187-203.

Résumé

Let $\Omega_1,\Omega_2\subset\mathbb R^\nu$ be compact sets. In the Hilbert space $L_2(\Omega_1\times\Omega_2)$, we study the spectral properties of selfadjoint partially integral operators $T_1$, $T_2$, and $T_1+T_2$, with \begin{align*} (T_1 f)(x,y)=\int_{\Omega_1}k_1(x,s,y)f(s,y)d\mu(s), \\ (T_2 f)(x,y)=\int_{\Omega_2}k_2(x,t,y)f(x,t)d\mu(t), \end{align*} whose kernels depend on three variables. We prove a theorem describing properties of the essential and discrete spectra of the partially integral operator $T_1+T_2$.

Bibliographie

[1] Aleksandrov V. M., Kovalenko E. V., Zadachi mekhaniki sploshnykh sred so smeshannymi granichnymi usloviyami, Nauka, M., 1986 | MR

[2] Kakichev V. A., Kovalenko N. V., “K teorii dvumernykh integralnykh uravnenii s chastnymi integralami”, Ukr. mat. zhurn., 25:3 (1973), 302–312 | Zbl

[3] Kalitvin A. S., “O spektre i sobstvennykh funktsiyakh ChIO tipa V. I. Romanovskogo”, Funktsionalnyi analiz, 22, Ulyanovsk. gos. ped. inst., Ulyanovsk, 1984, 35–45

[4] Kalitvin A. S., “O multispektre lineinykh operatorov”, Operatory i ikh prilozheniya. Priblizheniya funktsii. Uravneniya, Leningrad. gos. ped. inst., L., 1985, 91–99 | MR

[5] Kalitvin A. S., “O spektre nekotorykh klassov operatorov s chastnymi integralami”, Operatory i ikh prilozheniya. Priblizheniya funktsii. Uravneniya, Leningrad. gos. ped. inst., L., 1985, 27–35 | MR

[6] Kalitvin A. S., Issledovanie po ChIO, Dis. ... kand. fiz.-matem. nauk, Leningrad. ped. inst., L., 1986

[7] Kalitvin A. S., “O spektre lineinykh operatorov s chastnymi integralami i polozhitelnymi yadrami”, Operatory i ikh prilozheniya. Approksimatsiya funktsii. Spektralnaya teoriya, Leningrad. gos. ped. inst., L., 1988, 43–50 | MR | Zbl

[8] Kalitvin A. S., “O nekotorom klasse chastichno integralnykh uravnenii v aerodinamike”, Sostoyanie i perspektiva razvitiya nauki i tekhniki pod Lipetskom, 1994, 210–212

[9] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, T. 1, Mir, M., 1977 | MR

[10] Smirnov V. I., Kurs vysshei matematiki, T. 4, Ch. I, Nauka, M., 1974 | MR

[11] Fridrikhs K. O., Vozmuschenie spektra operatorov v gilbertovom prostranstve, Mir, M., 1969

[12] Shabat B. V., Vvedenie v kompleksnyi analiz, Ch. I, Nauka, M., 1985 | MR

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

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

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

[16] Aleksandrov V. M., Kovalenko E. V., “On some class of integral equations arising in mixed boundary value problems of continuum mechanics”, Soviet Phys. Dokl., 25:2 (1980), 354–356 | Zbl

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

[18] Mogilner A. I., “Hamiltonians in solid-state physics as multiparticle discrete Schrödinger operators: problems and results”, Adv. Soviet Math., 5, Amer. Math. Soc., Providence, RI, 1991, 139–194 | MR