RUS  ENG
Full version
JOURNALS // Uspekhi Matematicheskikh Nauk
Uspekhi Mat. Nauk, 1989, Volume 44, Issue 2(266), Pages 121–184 (Mi rm2275)
Spectral theory of two-dimensional periodic operators and its applications
I. M. Krichever

References
1. Dubrovin B. A., Matveev V. B., Novikov S. P., “Nelineinye uravneniya tipa Kortevega-de Friza, konechnozonnye lineinye operatory i abelevy mnogoobraziya”, UMN, 31:1 (1976), 55–136  mathnet  mathscinet  zmath
2. Zakharov V. E., Manakov S. V., Novikov S. P., Pitaevskii L. P., Teoriya solitonov: Metod obratnoi zadachi, Nauka, M., 1980  mathscinet
3. McKean H., van Moerbeke P., “The spectrum of Hill's equation”, Invent. Math., 30 (1975), 217–274  crossref  mathscinet  zmath  adsnasa
4. Lax P. D., “Periodic solutions of Korteweg-de Vries equation”, Comm. Pure and Appl. Math., 28 (1975), 141–188  mathscinet  zmath
5. McKean H., Trubovitz E., “Hill's operator and hyperelliptic functions theory in the presence of infinetely many branch points”, Comm. Pure and Appl. Math., 29 (1977), 143–226  crossref  mathscinet
6. Krichever I. M., “Algebro-geometricheskoe postroenie uravnenii Zakharova-Shabata i ikh periodicheskikh reshenii”, DAN SSSR, 227:2 (1976)  zmath
7. Krichever I. M., “Integrirovanie nelineinykh uravnenii metodami algebraicheskoi geometrii”, Funktsion. analiz i ego pril., 11:1 (1977), 15–31  mathnet  zmath
8. Krichever I. M., “Metody algebraicheskoi geometrii v teorii nelineinykh uravnenii”, UMN, 32:6 (1977), 180–208  mathnet
9. Dubrovin B. A., “Teta-funktsii i nelineinye uravneniya”, UMN, 36:2 (1981), 11–80  mathnet  mathscinet  zmath
10. Krichever I. M., Novikov S. P., “Golomorfnye rassloeniya nad algebraicheskimi krivymi i nelineinye uravneniya”, UMN, 35:6 (1980), 47–68  mathnet  mathscinet  zmath
11. Krichever I. M., “Nelineinye uravneniya i ellipticheskie krivye”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 23, VINITI, M., 1983, 79–136  mathnet  mathscinet  zmath
12. Dubrovin B. A., “Matrichnye konechnozonnye operatory”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 23, VINITI, M., 1983, 33–78  mathnet  mathscinet  zmath
13. Novikov S. P., “Dvumernye operatory Shredingera v periodicheskikh polyakh”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 23, VINITI, M., 1983, 3–32  mathnet
14. Dubrovin B. A., Krichever I. M., Novikov S. P., “Integriruemye sistemy, I”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Fundamentalnye napravleniya, 4, VINITI, M., 1985, 179–285  mathnet  mathscinet
15. Zakharov V. E., Shabat A. B., “Integrirovanie uravnenii matematicheskoi fiziki metodom obratnoi zadachi teorii rasseyaniya, I”, Funktsion. analiz i ego pril., 8:3 (1974), 43–53  mathnet  mathscinet  zmath
16. Dryuma V. S., “Ob analiticheskom reshenii dvumernogo uravneniya Kortevega-de Friza”, Pisma v ZhETF, 19:12 (1974), 219–225
17. Zakharov V. E., Shulman E. I., “Voprosy integriruemosti prostranstvenno-dvumernykh sistem”, DAN SSSR, 283:6 (1985), 1325–1328  mathnet  mathscinet
18. Krichever I. M., “Periodicheskaya zadacha dlya uravneniya KP-2”, DAN SSSR, 298:4 (1988)  mathscinet  zmath
19. Krichever I. M., Grinevich P. G., “Algebraic Geometry Methods in soliton theory”, Physica D, Nonlinear phenomena, University Press, Manchester, 1988  mathscinet
20. Dobrokhotov S. Yu., Maslov V. P., “Konechnozonnye pochti periodicheskie resheniya v VKB-priblizheniyakh”, Itogi nauki i tekhniki. Sovremennye problemy matematiki, 15, VINITI, M., 1980, 3–94  mathnet  mathscinet
21. Dobrokhotov S. Y., Maslov V. P., “Multiphase asymptotics of non-linear partial differential equations with a small parameter”, Sov. Scien. Reviews. Math. Phys. Rev., 3 (1982), 221–280  mathscinet
22. Krichever I. M., “Metod usredneniya dlya dvumernykh «integriruemykh» uravnenii”, Funktsion. analiz i ego pril., 22:3 (1988), 37–52  mathnet  mathscinet  zmath
23. Tsarev S. P., “O skobkakh Puassona i odnomernykh gamiltonovykh sistemakh gidrodinamicheskogo tipa”, DAN SSSR, 282:3 (1985), 534–537  mathnet  mathscinet
24. Dubrovin B. A., Novikov S. P., “Gamiltonov formalizm odnomernykh sistem gidrodinamicheskogo tipa i metod usredneniya Bogolyubova-Uizema”, DAN SSSR, 270:3 (1983), 781–785  mathnet  mathscinet  zmath
25. Dubrovin B. A., Novikov S. P., “O skobkakh Puassona gidrodinamicheskogo tipa”, DAN SSSR, 279:2 (1984), 294–297  mathnet  mathscinet  zmath
26. Bakhvalov N. S., Zhileikin Ya. M., Zabolotskaya E. A., Nelineinaya teoriya zvukovykh puchkov, Nauka, M., 1982  mathscinet  zmath
27. Dubrovin B. A., Krichever I. M., Novikov S. P., “Uravnenie Shredingera v periodicheskom pole i rimanovy poverkhnosti”, DAN SSSR, 229:1 (1976), 15–18  mathnet  mathscinet  zmath
28. Veselov A. P., Novikov S. P., “Konechnozonnye dvumernye periodicheskie operatory Shredingera: yavnye formuly i evolyutsionnye uravneniya”, DAN SSSR, 279:1 (1984), 20–24  mathnet  mathscinet  zmath
29. Veselov A. P., Novikov S. P., “Konechnozonnye dvumernye periodicheskie operatory Shredingera: potentsialnyi sluchai”, DAN SSSR, 279:4 (1984), 784–788  mathnet  mathscinet  zmath
30. Kuchment P. A., “Teoriya Floke dlya differentsialnykh uravnenii v chastnykh proizvodnykh”, UMN, 37:4 (1982), 3–52  mathnet  mathscinet  zmath
31. Novikov S. P., “Periodicheskaya zadacha dlya uravneniya Kortevega-de Friza”, Funktsion. analiz i ego pril., 8:3 (1974), 54–66  mathnet  mathscinet  zmath
32. Burchnall J. L., Chaundy T. W., “Commutative ordinary differential operators, I”, Proc. London Math. Soc., 21 (1922), 420–440  crossref
33. Burchnall J. L., Chaundy T. W., “Commutative ordinary differential operators, II”, Proc. London Math. Soc., 118 (1928), 557–583  crossref  zmath  adsnasa
34. Baker H. F., “Note on the foregoing paper “Commutative ordinary differential operators””, Proc. Royal Soc. London, 118 (1928), 584–593  crossref  zmath  adsnasa
35. Baker H. F., Abelian functions, Cambridge, 1897
36. Drinfeld V. G., “O kommutativnykh podkoltsakh nekotorykh nekommutativnykh kolets”, Funktsion. analiz i ego pril., 11:1 (1977), 11–15  mathnet  mathscinet
37. Krichever I. M., “Kommutativnye koltsa obyknovennykh lineinykh differentsialnykh operatorov”, Funktsion. analiz i ego pril., 12:3 (1978), 20–31  mathnet  mathscinet  zmath
38. Mumford D., “An algebro-geometrical construction of commutative operators and of solutions to the Toda lattice, $\mathrm{KdV}$ equation and related non-linear equations”, Symp. Alg. Geom. (Kyoto, 1977), Kinokunia–Tokyo, 1978  zmath
39. Tyurin A. N., “Klassifikatsiya vektornykh rassloenii nad algebraicheskoi krivoi proizvolnogo roda”, Izv. AN SSSR. Ser. mat., 29 (1965), 657–688  mathnet  mathscinet  zmath
40. Rodin Yu. L., “Kraevaya zadacha Rimana dlya differentsialov na rimanovykh poverkhnostyakh”, Uch. zap. Perm. un-ta, 17:2 (1980), 83–85  mathscinet
41. Koppelman W., “Singular integral equations, boundary value problem and Riemann-Roch theorem”, J. Math. and Mech., 10:2 (1961), 247–277  mathscinet  zmath
42. Krichever I. M., Novikov S. P., “Golomorfnye rassloeniya nad rimanovymi poverkhnostyami i uravnenie Kadomtseva-Petviashvili”, Funktsion. analiz i ego pril., 12:4 (1978), 41–52  mathnet  mathscinet  zmath
43. Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, Nauka, M., 1967  mathscinet
44. Grinevich P. G., Novikov S. P., “O spektralnoi teorii kommutiruyuschikh operatorov ranga 2 s periodicheskimi koeffitsientami”, Funktsion. analiz i ego pril., 16:1 (1982), 25–26  mathnet  mathscinet  zmath
45. Krichever I. M., Novikov S. P., “Golomorfnye rassloeniya i nelineinye uravneniya. Konechnozonnye resheniya ranga 2”, DAN SSSR, 247:1 (1979)  mathscinet  zmath
46. Sato M., Sato Y., “Soliton equations as dynamical systems on infinite dimensional Grassmann manifold”, Non-linear partial differential equations in applied science, North Holland, Amsterdam, New York, 1983, 259–271  mathscinet  zmath
47. Segal G., Wilson G., “Loop groups and equations of $\mathrm{KdV}$ type”, IHES Publ. Math., 61 (1985), 5–65  mathscinet  zmath
48. Krichever I. M., “Metod Laplasa, algebraicheskie krivye i nelineinye uravneniya”, Funktsion. analiz i ego pril., 18:3 (1984), 43–56  mathnet  mathscinet  zmath
49. Krichever I. M., Novikov S. P., “Algebry tipa Virasoro, rimanovy poverkhnosti i struktury teorii solitonov”, Funktsion. analiz i ego pril., 21:2 (1987), 46–63  mathnet  mathscinet  zmath
50. Dubrovin B. A., “Uravnenie Kadomtseva–Petviashvili i sootnosheniya mezhdu periodami golomorfnykh differentsialov na rimanovykh poverkhnostyakh”, Izv. AN SSSR. Ser. mat., 45:5 (1981), 1015–1028  mathnet  mathscinet  zmath
51. Shiota T., “Characterization of Jacobian varieties in terms of soliton equations”, Invent. Math., 83 (1986), 333–382  crossref  mathscinet  zmath  adsnasa
52. Krichever I. M., “Spektralnaya teoriya «konechnozonnykh» nestatsionarnykh operatorov Shredingera. Nestatsionarnaya model Paierlsa”, Funktsion. analiz i ego pril., 20:3 (1986), 42–54  mathnet  mathscinet  zmath
53. Marchenko V. A., Operatory Shturma-Liuvilya i ikh prilozheniya, Naukova dumka, Kiev, 1977  mathscinet
54. Firsova N. E., “Rimanova poverkhnost kvaziimpulsa i teoriya rasseyaniya dlya vozmuschennogo operatora Khilla”, 3ap. nauch. sem. LOMI, 51 (1975), 183–196  mathnet  mathscinet  zmath
55. Its A. R., Matveev V. B., “Operatory Shredingera s konechnozonnym spektrom i $N$-solitonnye resheniya uravneniya Kortevega-de Friza”, Teoret. i mat. fizika, 23:1 (1975), 51–67  mathnet  mathscinet
56. Cherednik I. V., “Ellipticheskie krivye i matrichnye solitonnye differentsialnye uravneniya”, Itogi nauki i tekhniki. Algebra. Geometriya. Topologiya, 22, VINITI, 1984, 205–265  mathnet  mathscinet
57. Dubrovin B. A., Natanzon S. M., “Veschestvennye teta-funktsionalnye resheniya uravneniya Kadomtseva-Petviashvili”, Izv. AN SSSR. Ser. mat., 52:2 (1988), 267–286  mathnet  mathscinet  zmath
58. Fay J., Theta-functions on Riemann surfaces, Lect. Notes Math., 352, 1973  mathscinet  zmath
59. Flashka H., Forest M. G., McLaughlin D. W., “Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation”, Comm. Pure Appl. Math., 33:6 (1980), 739–794  crossref  mathscinet  adsnasa
60. Zakharov V. E., “Uravneniya Benni i kvaziklassicheskoe priblizhenie v metode obratnoi zadachi”, Funktsion. analiz i ego pril., 14:2 (1980), 15–24  mathnet  mathscinet  zmath
61. Geogdzhaev V. V., “Reshenie uravnenii Benni metodom obratnoi zadachi rasseyaniya”, Teoret. i mat. fiz., 73:2 (1988), 255–263  mathnet  mathscinet  zmath
62. Gurevich L. P., Pitaevskii L. P., “Nestatsionarnaya struktura besstolknovitelnoi udarnoi volny”, ZhETF, 65:3 (1973), 590–604  adsnasa
63. Avilov V. P., Krichever I. M., Novikov S. P., “Evolyutsiya uitemovskoi zony v teorii Kortevega-de Friza”, DAN SSSR, 295:2 (1987)  mathscinet  zmath
64. Natanzon S. M., “Nesingulyarnye konechnozonnye dvumernye operatory Shredingera i primiany veschestvennykh krivykh”, Funktsion. analiz i ego pril., 22:1 (1988), 79–80  mathnet  mathscinet  zmath
65. Taimanov I. K., “Blokhovskie sobstvennye funktsii dlya nekotorykh dvumernykh periodicheskikh lineinykh operatorov”, DAN SSSR, 289:5 (1986)
66. Veselov A. P., Krichever I. M., Novikov S. P., “Two-dimensional periodic Schrödinger operators and Prym's theta-functions”, Geom. Today Int. Conf. (Rome, June 4–11, 1984), Boston, 1985, 283–301  mathscinet  zmath
67. Mumford D., Tata Lectures on Theta, I; II, Boston, 1983  mathscinet; Basel, 1984; Mamford D., Lektsii o teta-funktsiyakh, Mir, M., 1988  mathscinet
68. Date E., Jimbo M., Kashivara, Miva T., “Transformation groups for soliton equations, I”, Proc. Japan Acad., 57 A (1981), 342–347  crossref  mathscinet

© Steklov Math. Inst. of RAS, 2026