Geodesic
Analytic perturbation theory for a~periodic potential
Izvestiya. Mathematics , Tome 34 (1990) no. 1, pp. 43-64.

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

The operator $\mathbf H_\alpha=(-\Delta)^l+\alpha V$ is considered in $L_2(\mathbf R^n)$; here $4l>n+1$, $n\geqslant2$, $V$ is a periodic potential, and $\alpha$ is a perturbation parameter, $-1\leqslant\alpha\leqslant1$. An analytic perturbation theory with respect to $\alpha$ is constructed for Block eigenfunctions and the corresponding eigenvalues of $\mathbf H_\alpha$. It is proved that, for large energies, when the quasimomentum belongs to a sufficiently rich set they admit expansion in a Taylor series in the disk $|\alpha|\leqslant1$, and these series are asymptotic in the energy and infinitely differentiable with respect to the quasimomentum. Bibliography: 14 titles. 
@article{IM2_1990_34_1_a2,
     author = {Yu. E. Karpeshina},
     title = {Analytic perturbation theory for a~periodic potential},
     journal = {Izvestiya. Mathematics },
     pages = {43--64},
     publisher = {mathdoc},
     volume = {34},
     number = {1},
     year = {1990},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1990_34_1_a2/}
}
TY  - JOUR
AU  - Yu. E. Karpeshina
TI  - Analytic perturbation theory for a~periodic potential
JO  - Izvestiya. Mathematics 
PY  - 1990
SP  - 43
EP  - 64
VL  - 34
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_1990_34_1_a2/
LA  - en
ID  - IM2_1990_34_1_a2
ER  - 
%0 Journal Article
%A Yu. E. Karpeshina
%T Analytic perturbation theory for a~periodic potential
%J Izvestiya. Mathematics 
%D 1990
%P 43-64
%V 34
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_1990_34_1_a2/
%G en
%F IM2_1990_34_1_a2
Yu. E. Karpeshina. Analytic perturbation theory for a~periodic potential. Izvestiya. Mathematics , Tome 34 (1990) no. 1, pp. 43-64. http://geodesic.mathdoc.fr/item/IM2_1990_34_1_a2/

[1] Gelfand I. M., “Razlozhenie po sobstvennym funktsiyam uravneniya s periodicheskimi koeffitsientami”, Dokl. AN SSSR, 73:6 (1950), 1117–1120 | MR

[2] Gokhberg I. Ts., Krein M. G., Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov, Nauka, M., 1965

[3] Birman M. Sh., Solomyak M. Z., Spektralnaya teoriya samosopryazhennykh operatorov v gilbertovom prostranstve, Leningradskii un-t, L., 1980 | MR

[4] Skriganov M. M., “Dokazatelstvo gipotezy Bete–Zommerfelda v razmernosti dva”, Dokl. AN SSSR, 248:1 (1979), 39–42 | MR | Zbl

[5] Skriganov M. M., “O gipoteze Bete–Zommerfelda”, Dokl. AN SSSR, 244:3 (1979), 555–534 | MR

[6] Skriganov M. M., “Konechnost chisla lakun v spektre mnogomernogo poligarmonicheskogo operatora s periodicheskim potentsialom”, Matem. sb., 113(155):1 (1980), 133–146 | MR

[7] Popov V. N., Skriganov M. M., “Zamechanie o stroenii spektra dvumernogo operatora Shredingera s periodicheskim potentsialom”, Zap. nauch. seminarov LOMI, 109, 1981, 131–133 | MR | Zbl

[8] Skriganov M. M., Geometricheskie i arifmeticheskie metody v spektralnoi teorii mnogomernykh periodicheskikh operatorov, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 171, 1985 | MR | Zbl

[9] Skriganov M. M., “Mnogomernyi operator Shredingera s periodicheskim potentsialom”, Izv. AN SSSR. Ser. matem., 47:3 (1983), 659–687 | MR

[10] Skriganov M. M., Dokazatelstvo gipotezy Bete–Zommerfelda v razmernosti tri, Preprint, R-6-84, LOMI, L., 1984 | MR

[11] Veliev O. A., “O spektre operatora Shredingera s periodicheskim potentsialom”, Dokl. AN SSSR, 268:6 (1983), 1289–1292 | MR

[12] Veliev O. A., Molchanov S. A., “Stroenie spektra periodicheskogo operatora Shredingera na evklidovom tore”, Funkts. analiz i ego prilozh., 19:3 (1985), 86–87 | MR | Zbl

[13] Veliev O. A., Asimptoticheskie formuly dlya sobstvennykh chisel mnogomernogo operatora Shredingera i periodicheskie differentsialnye operatory, Preprint No 157, In-t Fiziki AN AzerSSR, Baku, 1985 | Zbl

[14] Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | Zbl