Geodesic
On the Spectrum of Degenerate Operator Equations
Matematičeskie zametki, Tome 68 (2000) no. 5, pp. 677-691 
V. V. Kornienko. On the Spectrum of Degenerate Operator Equations. Matematičeskie zametki, Tome 68 (2000) no. 5, pp. 677-691. http://geodesic.mathdoc.fr/item/MZM_2000_68_5_a4/
@article{MZM_2000_68_5_a4,
     author = {V. V. Kornienko},
     title = {On the {Spectrum} of {Degenerate} {Operator} {Equations}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {677--691},
     year = {2000},
     volume = {68},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2000_68_5_a4/}
}
TY  - JOUR
AU  - V. V. Kornienko
TI  - On the Spectrum of Degenerate Operator Equations
JO  - Matematičeskie zametki
PY  - 2000
SP  - 677
EP  - 691
VL  - 68
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/MZM_2000_68_5_a4/
LA  - ru
ID  - MZM_2000_68_5_a4
ER  - 
%0 Journal Article
%A V. V. Kornienko
%T On the Spectrum of Degenerate Operator Equations
%J Matematičeskie zametki
%D 2000
%P 677-691
%V 68
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_2000_68_5_a4/
%G ru
%F MZM_2000_68_5_a4

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

We study the distribution in the complex plane $\mathbb C$ of the spectrum of the operator $L=L(\alpha,a,A)$, $\alpha\in\mathbb R$, $a\in\mathbb C$, generated by the closure in $H=\mathscr L_2(0,b)\otimes\mathfrak H$ of the operation $t^\alpha aD_t^2+A$ originally defined on smooth functions $u(t)\colon[0,b]\to\mathfrak H$ with values in a Hilbert space $\mathfrak H$ satisfying the Dirichlet conditions $u(0)=u(b)=0$. Here $D_t\equiv d/dt$ and $A$ is a model operator acting in $\mathfrak H$. Criterial conditions on the parameter $\alpha$ for the eigenfunctions of the operator $L\colon H\to H$ to form a complete and minimal system as well as a Riesz basis in the Hilbert space $H$ are given.

[1] Sobolev S. L., Nekotorye prilozheniya funktsionalnogo analiza v matematicheskoi fizike, Izd-vo LGU, L., 1955

[2] Khille E., Fillips R., Funktsionalnyi analiz i polugruppy, IL, M., 1962

[3] Dezin A. A., Obschie voprosy teorii granichnykh zadach, Nauka, M., 1980 | Zbl

[4] Dezin A. A., “O slaboi i silnoi irregulyarnosti”, Differents. uravneniya, 17:10 (1981), 1851–1858 | MR | Zbl

[5] Romanko V. K., “O sobstvennykh znacheniyakh kraevykh zadach dlya nekotorykh uravnenii, menyayuschikh tip”, Differents. uravneniya, 19:10 (1983), 1759–1764 | MR | Zbl

[6] Kornienko V. V., International Conference on Some Topics of Mathematics, Abstracts, Samarkand, 1996, 84

[7] Dezin A. A., “Vyrozhdayuschiesya operatornye uravneniya”, Matem. sb., 115 (157):3 (7) (1981), 323–336 | MR | Zbl

[8] Tepoyan L. P., Vyrozhdayuschiesya differentsialno-operatornye uravneniya chetvertogo poryadka, Diss. ... k.f.-m.n., M., 1986

[9] Stepanov V. V., Kurs differentsialnykh uravnenii, Gostekhizdat, M., 1945

[10] Smirnov V. I., Kurs vysshei matematiki, T. 3, Ch. 2, Nauka, M., 1974

[11] Naimark M. A., Lieinye differentsialnye operatory, Nauka, M., 1969

[12] Kachmazh S., Shteingauz G., Teoriya ortogonalnykh ryadov, GIFML, M., 1958

[13] Keldysh M. V., “O nekotorykh sluchayakh vyrozhdeniya uravnenii ellipticheskogo tipa na granitse oblasti”, Dokl. AN SSSR, 77:2 (1951), 181–183 | MR

[14] Kornienko V. V., “K slaboi i silnoi irregulyarnosti”, Sib. matem. zh., 37:3 (1996), 599–609 | MR | Zbl