Geodesic
Spectral properties of two classes of periodic difference operators
Sbornik. Mathematics, Tome 193 (2002) no. 4, pp. 559-584 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A study is made of the iso-energetic spectral problem for two classes of multidimensional periodic difference operators. The first class of operators is defined on a regular simplicial lattice. The second class is defined on a standard rectangular lattice and is the difference analogue of a multidimensional Schrödinger operator. The varieties arising in the direct spectral problem are described, along with the divisor of an eigenfunction, defined on the spectral variety, of the corresponding operator. Multidimensional analogues are given for the Veselov–Novikov correspondences connecting the divisors of the eigenfunction with the canonical divisor of the spectral variety. Also, a method is proposed for solving the inverse spectral problem in terms of $\theta$-functions of curves lying “at infinity” on the spectral variety. 
@article{SM_2002_193_4_a4,
     author = {A. A. Oblomkov},
     title = {Spectral properties of two classes of periodic difference operators},
     journal = {Sbornik. Mathematics},
     pages = {559--584},
     year = {2002},
     volume = {193},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2002_193_4_a4/}
}
TY  - JOUR
AU  - A. A. Oblomkov
TI  - Spectral properties of two classes of periodic difference operators
JO  - Sbornik. Mathematics
PY  - 2002
SP  - 559
EP  - 584
VL  - 193
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_2002_193_4_a4/
LA  - en
ID  - SM_2002_193_4_a4
ER  - 
%0 Journal Article
%A A. A. Oblomkov
%T Spectral properties of two classes of periodic difference operators
%J Sbornik. Mathematics
%D 2002
%P 559-584
%V 193
%N 4
%U http://geodesic.mathdoc.fr/item/SM_2002_193_4_a4/
%G en
%F SM_2002_193_4_a4
A. A. Oblomkov. Spectral properties of two classes of periodic difference operators. Sbornik. Mathematics, Tome 193 (2002) no. 4, pp. 559-584. http://geodesic.mathdoc.fr/item/SM_2002_193_4_a4/

[1] Shadan K., Sabate P., Obratnye zadachi v kvantovoi teorii rasseyaniya, Mir, M., 1980 | MR

[2] Dubrovin A. B., Krichever I. M., Novikov S. P., “Uravnenie Shredingera v periodicheskom pole i rimanovy poverkhnosti”, Dokl. AN SSSR, 229:1 (1976), 15–18 | MR | Zbl

[3] Novikov S. P., Veselov A. P., “Two-dimensional Schrödinger operator: inverse scatering problem and evolutional equations”, Phis. D, 18 (1986), 267–273 | DOI | MR | Zbl

[4] Veselov A. P., Novikov S. P., “Konechnozonnye dvumernye operatory Shredingera. Potentsialnye operatory”, Dokl. AN SSSR, 1984:4 (1984), 784–788 | MR | Zbl

[5] Novikov S. P., Veselov A. P., “Exactly solvable two-dimensional Schrödinger operator and Laplace transformation”, Solitons, geometry, and topology: on the crossroad, Amer. Math. Soc. Transl. Ser. 2, 179 (33), Amer. Math. Soc., Providence, RI, 1997, 109–132 | MR | Zbl

[6] Krichever I. M., “Dvumernye periodicheskie raznostnye operatory i algebraicheskaya geometriya”, Dokl. AN SSSR, 285:1 (1985), 31–36 | MR | Zbl

[7] Veselov A. P., Krichever I. M., Novikov S. P., “Two-dimensional periodic Schrödinger operators and Prym's $\theta$-functions”, Geometry today, Internat. Conf. (Rome, 1984), Progr. Math., 60, 1985, 283–301 | MR | Zbl

[8] Oblomkov A. A., “O raznostnykh operatorakh na dvumernykh pravilnykh reshetkakh”, TMF, 127:1 (2001), 34–46 | MR | Zbl

[9] Oblomkov A. A., Penskoi A. V., “Two-dimensional algebro-geometric operators”, J. Phys. A, 33 (2000), 9255–9264 | DOI | MR | Zbl

[10] Novikov S. P., Dynnikov I. A., “Diskretnye simmetrii malomernykh operatorov i raznostnykh operatorov na pravilnykh reshetkakh i dvumernykh mnogoobraziyakh”, UMN, 52:5 (1997), 175–234 | MR | Zbl

[11] Oblomkov A. A., “Izoenergeticheskaya spektralnaya zadacha dlya mnogomernykh raznostnykh operatorov”, Funkts. analiz i ego prilozh. (to appear)

[12] Khovanskii A. G., “Mnogogranniki Nyutona (razreshenie osobennostei)”, Itogi nauki i tekhn. Sovrem. probl. matem., 22, VINITI, M., 1983, 207–239 | MR

[13] Danilov V. I., “Geometriya toricheskikh mnogoobrazii”, UMN, 33:2 (1978), 85–134 | MR | Zbl