Geodesic
Perturbation theory formulas for the Schrödinger equation with a nonsmooth periodic potential
Sbornik. Mathematics, Tome 71 (1992) no. 1, pp. 101-123 
Yu. E. Karpeshina. Perturbation theory formulas for the Schrödinger equation with a nonsmooth periodic potential. Sbornik. Mathematics, Tome 71 (1992) no. 1, pp. 101-123. http://geodesic.mathdoc.fr/item/SM_1992_71_1_a7/
@article{SM_1992_71_1_a7,
     author = {Yu. E. Karpeshina},
     title = {Perturbation theory formulas for the {Schr\"odinger} equation with a~nonsmooth periodic potential},
     journal = {Sbornik. Mathematics},
     pages = {101--123},
     year = {1992},
     volume = {71},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1992_71_1_a7/}
}
TY  - JOUR
AU  - Yu. E. Karpeshina
TI  - Perturbation theory formulas for the Schrödinger equation with a nonsmooth periodic potential
JO  - Sbornik. Mathematics
PY  - 1992
SP  - 101
EP  - 123
VL  - 71
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1992_71_1_a7/
LA  - en
ID  - SM_1992_71_1_a7
ER  - 
%0 Journal Article
%A Yu. E. Karpeshina
%T Perturbation theory formulas for the Schrödinger equation with a nonsmooth periodic potential
%J Sbornik. Mathematics
%D 1992
%P 101-123
%V 71
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1992_71_1_a7/
%G en
%F SM_1992_71_1_a7

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

Series from perturbation theory are constructed for the Bloch eigenvalues and eigenfunctions for the periodic Schrödinger operator in $R^3$. An extensive set of quasimomenta on which the series converge is described. It is shown that the series have asymptotic character at high energies. They are infinitely differentiable with respect to the quasimomentum, and preserve their asymptotic character under such differentiation.

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

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

[3] Veliev O. A., “Asimptoticheskie formuly dlya sobstvennykh chisel periodicheskogo operatora Shredingera i gipoteza Bete–Zommerfelda”, Funktsion. analiz i ego pril., 21:2 (1987), 1–15 | MR | Zbl

[4] Karpeshina Yu. E., “Teoriya vozmuschenii dlya operatora Shredingera s periodicheskim potentsialom”, Tr. MIAN, 188, 1990, 88–116 | MR | Zbl

[5] Madelung O., Teoriya tverdogo tela, Nauka, M., 1985

[6] Karpeshina Yu. E., “Analiticheskaya teoriya vozmuschenii dlya periodicheskogo potentsiala”, Izv. AN SSSR. Ser. matem., 53:1 (1989), 45–65 | MR

[7] Karpeshina Yu. E., “Formuly teorii vozmuschenii dlya poligarmonicheskogo operatora s negladkim periodicheskim potentsialom”, Problemy matematicheskoi fiziki, vyp. 13, Izd-vo LGU, L., 1989, 47–67