Geodesic
Eigenfunction expansions of the magnetic Schrödinger operator in bounded domains
Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 69 (2021), pp. 5-14 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this work, we introduce the magnetic Schrödinger operator corresponding to the generalized Dirichlet problem. We prove its self-adjointness and discreteness of the spectrum in bounded domains in the multidimensional case. We also prove the basis property of its eigenfunctions in the Lebesgue space and in the magnetic Sobolev space. We give a new characteristic of the definition domain of the magnetic Schrödinger operator. We investigate the existence and uniqueness of a solution of the magnetic Schrödinger equation with a spectral parameter. It is proved that if the spectral parameter is different from the eigenvalues, then the first generalized Dirichlet problem has a unique solution. We then find the solvability condition for the generalized Dirichlet problem when the spectral parameter coincides with the eigenvalue of the Schrödinger magnetic operator.
Keywords: magnetic Schrödinger operator, discrete spectrum, eigenvalues and eigenfunctions, eigenfunction expansions, theorems for existence and uniqueness of solutions. 
@article{VTGU_2021_69_a0,
     author = {A. R. Aliev and Sh. Sh. Radzhabov},
     title = {Eigenfunction expansions of the magnetic {Schr\"odinger} operator in bounded domains},
     journal = {Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika},
     pages = {5--14},
     year = {2021},
     number = {69},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTGU_2021_69_a0/}
}
TY  - JOUR
AU  - A. R. Aliev
AU  - Sh. Sh. Radzhabov
TI  - Eigenfunction expansions of the magnetic Schrödinger operator in bounded domains
JO  - Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
PY  - 2021
SP  - 5
EP  - 14
IS  - 69
UR  - http://geodesic.mathdoc.fr/item/VTGU_2021_69_a0/
LA  - ru
ID  - VTGU_2021_69_a0
ER  - 
%0 Journal Article
%A A. R. Aliev
%A Sh. Sh. Radzhabov
%T Eigenfunction expansions of the magnetic Schrödinger operator in bounded domains
%J Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika
%D 2021
%P 5-14
%N 69
%U http://geodesic.mathdoc.fr/item/VTGU_2021_69_a0/
%G ru
%F VTGU_2021_69_a0
A. R. Aliev; Sh. Sh. Radzhabov. Eigenfunction expansions of the magnetic Schrödinger operator in bounded domains. Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 69 (2021), pp. 5-14. http://geodesic.mathdoc.fr/item/VTGU_2021_69_a0/

[1] Krulikovskii N. N., Puti razvitiya spektralnoi teorii obyknovennykh differentsialnykh operatorov, TGU, Tomsk, 2008, 224 pp.

[2] Rajabov Sh.Sh., “Generalized Dirichlet problems for magnetic Schrodinger operator”, Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 45:1 (2019), 111–118 | MR | Zbl

[3] Leinfelder H., Simader C., “Schrodinger operators with singular magnetic vector potentials”, Math. Z., 176 (1981), 1–19 | DOI | MR | Zbl

[4] Balandin M. Yu., Shurina E. P., Vektornyi metod konechnykh elementov, NGTU, Novosibirsk, 2001, 69 pp.

[5] Gilbarg D., Trudinger N., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989, 464 pp.

[6] Raymond N., Elements of spectral theory, Master, France, 2017, 61 pp.

[7] Shubin M. A., Lektsii ob uravneniyakh matematicheskoi fiziki, MTsNMO, M., 2003, 303 pp.

[8] Rajabov Sh.Sh., “On the existence and uniqueness of the solution of Dirichlet generalized problem in arbitrary domain of n-dimensional space Rn for magnetic Schrodinger operator”, Azerbaijan Journal of Mathematics, 10:1 (2020), 172–180 | MR | Zbl

[9] Mizokhata S., Teoriya uravnenii s chastgnymi proizvodnymi, Mir, M., 1977, 504 pp.

[10] Adams R. A., Fournier J. J., Sobolev Spaces, Academic Press, Amsterdam–Boston, 2003, 317 pp. | MR | Zbl

[11] Aliev A. R., Eivazov E. Kh., “O diskretnosti spektra magnitnogo operatora Shredingera”, Funktsionalnyi analiz i ego prilozheniya, 46:4 (2012), 83–85 | MR | Zbl

[12] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, v. 4, Analiz operatorov, Mir, M., 1982, 428 pp.

[13] Akhiezer N. I., Glazman I. M., Teoriya lineinykh operatorov v gilbertovom prostranstve, v. 1, Izd-vo Kharkovskogo universiteta, Vischa shkola, Kharkov, 1977, 318 pp. | MR