Scientific heritage of L. D. Faddeev. Survey of papers

L. A. Takhtajan   ; A. Yu. Alekseev   ; I. Ya. Aref'eva   ; M. A. Semenov-Tian-Shansky   ; E. K. Sklyanin   ; F. A. Smirnov   ; S. L. Shatashvili

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 6, pp. 977-1081

Résumé

This survey was written by students of L. D. Faddeev under the editorship of L. A. Takhtajan. Sections 1.1, 1.2, 2–4, and 6 were written by Takhtajan, §§ 1.3 and 1.4 by F. A. Smirnov, §§ 5.1 and 5.2 by E. K. Sklyanin, §§ 5.3–5.6 by Sklyanin, Smirnov, and Takhtajan, § 7.1 by M. A. Semenov-Tian-Shansky, §§ 7.2–7.6 by Takhtajan and S. L. Shatashvili, § 7.7 by A. Yu. Alekseev and Shatashvili, and § 8 by I. Ya. Aref'eva. Bibliography: 130 titles.

Détail

Keywords: Schrödinger operator, eigenfunction expansion, scattering theory, inverse scattering problem, Korteweg–de Vries equation, complete integrability, inverse scattering method, Yang–Baxter equation, quantum inverse problem method, algebraic Bethe ansatz, quantum groups, quantum dilogarithm, quantization of gauge fields, Faddeev–Popov ghosts
Mots-clés : quantum anomalies.

Bibliographie

[1] “Edinstvennost resheniya obratnoi zadachi rasseyaniya”, Vestn. LGU. Ser. matem. mekh. astron., 11:7 (1956), 126–130 | MR

[2] “O razlozhenii proizvolnykh funktsii po sobstvennym funktsiyam operatora Shredingera”, Vestn. LGU. Ser. matem. mekh. astron., 12:7 (1957), 164–172 | MR | Zbl

[3] “O vyrazhenii dlya sleda raznosti dvukh singulyarnykh differentsialnykh operatorov Shturma–Liuvillya”, Dokl. AN SSSR, 115:5 (1957), 878–881 | MR | Zbl

[4] “O dispersionnykh sootnosheniyakh nerelyativistskoi teorii rasseyaniya”, ZhETF, 35:2(8) (1958), 433–439 ; “Dispersion relations in non-relativistic scattering theory”, Soviet Physics. JETP, 8 (1959), 299–303 | MR | Zbl

[5] “O svyazi $S$-matritsy i potentsiala dlya odnomernogo operatora Shredingera”, Dokl. AN SSSR, 121:1 (1958), 63–66 ; “On the relation between the $S$-matrix and potential for the one-dimensional Schrödinger operator”, Soviet Phys. Dokl., 3 (1959), 747–751 | MR | Zbl

[6] “K teorii vozmuschenii nepreryvnogo spektra”, Dokl. AN SSSR, 120:6 (1958), 1187–1190 (with O. A. Ladyzhenskoi) | MR | Zbl

[7] “Obratnaya zadacha kvantovoi teorii rasseyaniya”, UMN, 14:4(88) (1959), 57–119 ; “The inverse problem in the quantum theory of scattering”, J. Math. Phys., 4 (1963), 72–104 | MR | Zbl | DOI | MR | Zbl

[8] “Teoriya rasseyaniya dlya sistemy iz trekh chastits”, ZhETF, 39:5 (1960), 1459–1467 ; “Scattering theory for a three-particle system”, Soviet Physics. JETP, 12 (1961), 1014–1019 | MR

[9] “O formulakh sledov dlya differentsialnogo singulyarnogo operatora Shturma–Liuvillya”, Dokl. AN SSSR, 132:1 (1960), 13–16 (with V. S. Buslaevym) | MR | Zbl

[10] “Stroenie rezolventy operatora Shredingera sistemy trekh chastits s parnym vzaimodeistviem”, Dokl. AN SSSR, 138:3 (1961), 565–567 | MR

[11] “Zamechanie ob uravnenii Shredingera s singulyarnym potentsialom”, Dokl. AN SSSR, 137:5 (1961), 1011–1014 (with F. A. Berezinym) | MR | Zbl

[12] “Stroenie rezolventy operatora Shredingera sistemy trekh chastits i zadacha rasseyaniya”, Dokl. AN SSSR, 145:2 (1962), 301–304 ; “The construction of the resolvent of the Schrödinger operator for a three-particle system, and the scattering problem”, Soviet Phys. Dokl., 7 (1963), 600–602 | MR

[13] “Matematicheskie voprosy kvantovoi teorii rasseyaniya dlya sistemy trekh chastits”, Tr. MIAN SSSR, 69, Izd-vo AN SSSR, M.–L., 1963, 3–122 ; Mathematical aspects of the three-body problem in the quantum scattering theory, Israel Program for Scientific Translations, Jerusalem; Daniel Davey Co., Inc., New York, 1965, iii+110 pp. | MR | Zbl | MR | Zbl

[14] “O razdelenii effektov samodeistviya i rasseyaniya po teorii vozmuschenii”, Dokl. AN SSSR, 152:3 (1963), 573–576 ; “Distinction between interaction and scattering effects in perturbation theory”, Soviet Phys. Dokl., 8 (1964), 881–883 | MR

[15] “O modeli Fridrikhsa v teorii vozmuschenii nepreryvnogo spektra”, Kraevye zadachi matematicheskoi fiziki. 2, Sbornik rabot. Posvyaschaetsya pamyati Vladimira Andreevicha Steklova v svyazi so stoletiem so dnya ego rozhdeniya, Tr. MIAN SSSR, 73, Nauka, M.–L., 1964, 292–313 ; “On the Friedrichs model in the theory of perturbations of a continuous spectrum”, Amer. Math. Soc. Transl. Ser. 2, 62, Amer. Math. Soc., Providence, RI, 1967, 177–203 | MR | Zbl | DOI

[16] “Svoistva $S$-matritsy odnomernogo uravneniya Shrëdingera”, Kraevye zadachi matematicheskoi fiziki. 2, Sbornik rabot. Posvyaschaetsya pamyati Vladimira Andreevicha Steklova v svyazi so stoletiem so dnya ego rozhdeniya, Tr. MIAN SSSR, 73, Nauka, M.–L., 1964, 314–336 ; “Properties of the $S$-matrix of the one-dimensional Schrödinger equation”, Amer. Math. Soc. Transl. Ser. 2, 65, Amer. Math. Soc., Providence, RI, 1967, 139–166 | MR | Zbl | DOI

[17] “Rastuschie resheniya uravneniya Shredingera”, Dokl. AN SSSR, 165:3 (1965), 514–517 ; “Increasing solutions of the Schrödinger equation”, Soviet Phys. Dokl., 10 (1965) (1966), 1033–1035 | Zbl

[18] “Faktorizatsiya $S$-matritsy mnogomernogo operatora Shredingera”, Dokl. AN SSSR, 167:1 (1966), 69–72 | MR

[19] “Razlozhenie po sobstvennym funktsiyam operatora Laplasa na fundamentalnoi oblasti diskretnoi gruppy na ploskosti Lobachevskogo”, Tr. MMO, 17, Izd-vo Mosk. un-ta, M., 1967, 323–350 | MR | Zbl

[20] Teoriya vozmuschenii dlya kalibrovochno-invariantnykh polei, Preprint ITF-67-036, ITF AN USSR, Kiev, 1967 (with V. N. Popovym), 28 pp.; Perturbation theory for gauge-invariant fields, Preprint NAL-THY-57, 1972 (with V. N. Popov), 36 pp.; 50 years of Yang–Mills theory, ed. G. t'Hooft, World Sci. Publ., Hackensack, NJ, 2005, 40–60 | DOI | MR | Zbl

[21] “Feynman diagrams for the Yang–Mills field”, Phys. Lett. B, 25:1 (1967), 29–30 (with V. N. Popov) | DOI

[22] “Integral Feinmana dlya singulyarnykh lagranzhianov”, TMF, 1:1 (1969), 3–18 | MR | Zbl

[23] “Asimptoticheskie usloviya i infrakrasnye raskhodimosti v kvantovoi elektrodinamike”, TMF, 4:2 (1970), 153–170 (with P. P. Kulishom) | Zbl

[24] “Uravnenie Kortevega–de Frisa – vpolne integriruemaya gamiltonova sistema”, Funkts. analiz i ego pril., 5:4 (1971), 18–27 (with V. E. Zakharovym) | MR | Zbl

[25] “Trekhmernaya obratnaya zadacha kvantovoi teorii rasseyaniya”, Obratnye zadachi dlya differentsialnykh uravnenii, Sb. tr. vsesoyuzn. simpoziuma (Novosibirsk, 1971), VTs SO AN SSSR, Novosibirsk, 1972, 14–30

[26] “Teoriya rasseyaniya i avtomorfnye funktsii”, Kraevye zadachi matematicheskoi fiziki i smezhnye voprosy teorii funktsii. 6, Zap. nauch. sem. LOMI, 27, Izd-vo “Nauka”, Leningr. otd., L., 1972, 161–193 (with B. S. Pavlovym) ; “Scattering theory and automorphic functions”, J. Soviet Math., 3:4 (1975), 522–548 (with B. S. Pavlov) | MR | Zbl | DOI

[27] “Nearifmeticheskii vyvod formuly sleda Selberga”, Differentsialnaya geometriya, gruppy Li i mekhanika, Zap. nauch. sem. LOMI, 37, Izd-vo “Nauka”, Leningr. otd., L., 1973, 5–42 (with A. B. Venkovym, V. L. Kalininym) ; “A non-arithmetic derivation of the Selberg trace formula”, J. Soviet Math., 8:2 (1977), 171–199 (with A. B. Venkov, V. L. Kalinin) | MR | Zbl | DOI

[28] “Obratnaya zadacha kvantovoi teorii rasseyaniya. II”, Itogi nauki i tekhn. Ser. Sovrem. probl. mat., 3, VINITI, M., 1974, 93–180 ; “Inverse problem of quantum scattering theory. II”, J. Soviet Math., 5:3 (1976), 334–396 | MR | Zbl | DOI

[29] “Polnoe opisanie reshenii ‘sine-Gordon’ uravneniya”, Dokl. AN SSSR, 219 (1974), 1334–1337 (with V. E. Zakharovym, L. A. Takhtadzhyanom) | MR | Zbl

[30] “Suschestvenno-nelineinaya odnomernaya model klassicheskoi teorii polya”, TMF, 21:2 (1974), 160–174 (with L. A. Takhtadzhyanom) ; “Essentially nonlinear one-dimensional model of classical field theory”, Theoret. and Math. Phys., 21:2 (1975), 1046–1057 (with L. A. Takhtadzhyan) | Zbl | DOI

[31] “Kvantovanie solitonov”, Pisma v ZhETF, 21:5 (1975), 302–305 (with V. E. Korepinym, P. P. Kulishom)

[32] “Kvantovanie solitonov”, TMF, 25:2 (1975), 147–163 (with V. E. Korepinym) | MR

[33] “Suschestvenno-nelineinaya odnomernaya model klassicheskoi teorii polya (Dopolnenie)”, TMF, 22:1 (1975), 143 (with L. A. Takhtadzhyanom)

[34] “Course 1. Introduction to functional methods”, Méthodes en théorie des champs – Methods in field theory (École d'Été Phys. Théor., Session XXVIII, Les Houches, 1975), North-Holland, Amsterdam, 1976, 1–40 | MR

[35] “About the zero mode problem in the quantization of solitons”, Phys. Lett. B, 63:4 (1976), 435–438 (with V. E. Korepin) | DOI

[36] “Sravnenie tochnykh kvantovykh i kvaziklassicheskikh otvetov dlya nelineinogo uravneniya Shredingera”, TMF, 28:1 (1976), 38–45 (with P. P. Kulishom, S. V. Manakovym) | MR

[37] “Gamiltonova sistema, svyazannaya s uravneniem $u_{\xi_\eta}+\sin u=0$”, Teoriya chisel, matematicheskii analiz i ikh prilozheniya, Sbornik statei. Posvyaschaetsya akademiku Ivanu Matveevichu Vinogradovu k ego vosmidesyatipyatiletiyu, Tr. MIAN SSSR, 142, 1976, 254–266 (with L. A. Takhtadzhyanom) ; “The Hamiltonian system connected with the equation ${u_{\xi_\eta}+\sin u=0}$”, Proc. Steklov Inst. Math., 142 (1979), 277–289 (with L. A. Takhtadzhyan) | MR | Zbl

[38] “Quantum theory of solitons”, Phys. Rep., 42:1 (1978), 1–87 (with V. E. Korepin) | DOI | MR

[39] “Kvantovomekhanicheskii podkhod k vpolne integriruemym modelyam teorii polya”, Dokl. AN SSSR, 243:6 (1978), 1430–1433 (with E. K. Sklyaninym)

[40] “Kvantovyi metod obratnoi zadachi. I”, TMF, 40:2 (1979), 194–220 (with E. K. Sklyaninym, L. A. Takhtadzhyanom) | MR

[41] “Kvantovyi metod obratnoi zadachi i $XYZ$ model Geizenberga”, UMN, 34:5 (1979), 13–63 (with L. A. Takhtadzhyanom) | MR

[42] “Quantum completely integrable models in field theory”, Mathematical physics reviews, v. 1, Soviet Sci. Rev. Sect. C: Math. Phys. Rev., 1, Harwood Academic, Chur, 1980, 107–155 | MR | Zbl

[43] “Two-dimensional integrable models in quantum field theory”, Phys. Scripta, 24:5 (1981), 832–835 | DOI | MR | Zbl

[44] “Spektr i rasseyanie vozbuzhdenii v odnomernom izotropnom magnetike Geizenberga”, Differentsialnaya geometriya, gruppy Li i mekhanika. IV, Zap. nauch. sem. LOMI, 109, Izd-vo “Nauka”, Leningr. otd., L., 1981, 134–178 (with L. A. Takhtadzhyanom) ; “Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model”, J. Soviet Math., 24:2 (1984), 241–267 (with L. A. Takhtadzhyan) | MR | Zbl | DOI

[45] “What is the spin of a spin wave?”, Phys. Lett. A, 85:6-7 (1981), 375–377 (with L. A. Takhtajan) | DOI | MR

[46] “Gamiltonovy struktury dlya integriruemykh modelei teorii polya”, TMF, 56:3 (1983), 323–343 (with N. Yu. Reshetikhinym) | MR

[47] “Integrable models in $(1+1)$-dimensional quantum field theory”, Recent advances in field theory and statistical mechanics (Les Houches, 1982), North-Holland, Amsterdam, 1984, 561–608 | MR

[48] “Algebraicheskie i gamiltonovy metody v teorii neabelevykh anomalii”, TMF, 60:2 (1984), 206–217 (with S. L. Shatashvili) ; “Algebraic and Hamiltonian methods in the theory of non-Abelian anomalies”, Theoret. and Math. Phys., 60:2 (1985), 770–778 (with S. L. Shatashvili) | MR | DOI

[49] “Operator anomaly for the Gauss law”, Phys. Lett. B, 145:1-2 (1984), 81–84 | DOI | MR

[50] Kvantovaya teoriya rasseyaniya dlya sistem neskolkikh chastits, Nauka, M., 1985 (with S. P. Merkurevym), 399 pp. ; Quantum scattering theory for several particle systems, Math. Phys. Appl. Math., 11, Kluwer Acad. Publ., Dordrecht, 1993 (with S. P. Merkuriev), xiv+404 pp. | MR | Zbl | DOI | MR | Zbl

[51] “Realization of the Schwinger term in the Gauss law and the possibility of correct quantization of a theory with anomalies”, Phys. Lett. B, 167:2 (1986), 225–228 (with S. L. Shatashvili) | DOI

[52] Gamiltonov podkhod v teorii solitonov, Nauka, M., 1986 (with L. A. Takhtadzhyanom), 528 pp. ; Hamiltonian methods in the theory of solitons, Springer Ser. Soviet Math., Springer-Verlag, Berlin, 1987 (with L. A. Takhtajan), ix+592 pp. ; Classics Math., Springer, Berlin, 2007, xiv+592 pp. | MR | DOI | MR | Zbl | MR | Zbl

[53] “Liouville model on the lattice”, Field theory, quantum gravity and strings (Meudon/Paris, 1984/1985), Lecture Notes in Phys., 246, Springer, Berlin, 1986, 166–179 (with L. A. Takhtajan) | DOI | MR

[54] “30 let v matematicheskoi fizike”, Matematicheskaya fizika i kompleksnyi analiz, Sbornik obzornykh statei 4. K 50-letiyu Instituta, Tr. MIAN SSSR, 176, 1987, 4–29 ; “30 years in mathematical physics”, Proc. Steklov Inst. Math., 176 (1988), 3–28 | MR | Zbl

[55] “Vyvod anomalnogo kommutatora v formalizme funktsionalnogo integrala”, TMF, 73:2 (1987), 187–190 (with A. Yu. Alekseevym, Ya. Madaichikom, S. L. Shatashvili) | MR

[56] “Quantization of Lie groups and Lie algebras”, Algebraic analysis, v. I, Academic Press, Boston, MA, 1988, 129–139 (with N. Yu. Reshetikhin, L. A. Takhtadzhan) | MR | Zbl

[57] Vvedenie v kvantovuyu teoriyu kalibrovochnykh polei, 2-e izd., Nauka, M., 1988 (with A. A. Slavnovym), 270 pp. ; Gauge fields. Introduction to quantum theory, Frontiers Phys., 83, Addison-Wesley Publishing Co., Redwood City, CA, 1991 (with A. A. Slavnov), xviii+217 pp. | MR | Zbl | MR | Zbl

[58] “Quantization of symplectic orbits of compact Lie groups by means of the functional integral”, Dedicated to I. M. Gelfand on his 75th birthday, J. Geom. Phys., 5:3 (1988), 391–406 (with A. Alekseev, S. Shatashvili) | DOI | MR | Zbl

[59] “Kvantovanie grupp Li i algebr Li”, Algebra i analiz, 1:1 (1989), 178–206 (with N. Yu. Reshetikhinym, L. A. Takhtadzhyanom) ; “Quantization of Lie groups and Lie algebras”, Leningrad Math. J., 1 (1990), 193–225 (with N. Yu. Reshetikhin, L. A. Takhtadzhyan) | MR | Zbl

[60] “On the exchange matrix for WZNW model”, Comm. Math. Phys., 132:1 (1990), 131–138 | DOI | MR | Zbl

[61] “Lectures on quantum inverse scattering method”, Integrable systems, Nankai Lectures Math. Phys. (Tianjin, 1987), World Sci. Publ., Teaneck, NJ, 1990, 23–70 | MR

[62] “$(T^{\ast}G)_{t}$: a toy model for conformal field theory”, Comm. Math. Phys., 141:2 (1991), 413–422 (with A. Yu. Alekseev) | DOI | MR | Zbl

[63] “Hidden quantum groups inside Kac–Moody algebra”, Comm. Math. Phys., 149:2 (1992), 335–345 (with A. Alekseev, M. Semenov-Tian-Shansky) | DOI | MR | Zbl

[64] “Kvantovyi metod obratnoi zadachi na diskretnom prostranstve-vremeni”, TMF, 92:2 (1992), 207–214 (with A. Yu. Volkovym) | MR

[65] “Abelian current algebra and the Virasoro algebra on the lattice”, Phys. Lett. B, 315:3-4 (1993), 311–318 (with A. Yu. Volkov) | DOI | MR | Zbl

[66] “Quantum dilogarithm”, Modern. Phys. Lett. A, 9:5 (1994), 427–434 (with R. M. Kashaev) | DOI | MR | Zbl

[67] “Hirota equation as an example of an integrable symplectic map”, Lett. Math. Phys., 32:2 (1994), 125–135 (with A. Yu. Volkov) | DOI | MR | Zbl

[68] “Algebraic aspects of the Bethe ansatz”, Internat. J. Modern Phys. A, 10:13 (1995), 1845–1878 | DOI | MR | Zbl

[69] “Instructive history of the quantum inverse scattering method”, KdV{'}95 (Amsterdam, 1995), Acta Appl. Math., 39:1-3 (1995), 69–84 | DOI | MR | Zbl

[70] “Discrete Heisenberg–Weyl group and modular group”, Lett. Math. Phys., 34:3 (1995), 249–254 | DOI | MR | Zbl

[71] “High-energy QCD as a completely integrable model”, Phys. Lett. B, 342:1-4 (1995), 311–322 (with G. P. Korchemsky) | DOI

[72] “How the algebraic Bethe ansatz works for integrable models”, Symétries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998, 149–219 | MR | Zbl

[73] “Modular double of a quantum group”, Conférence Moshé Flato 1999: Quantization, deformation, and symmetries (Dijon), v. 1, Math. Phys. Stud., 21, Kluwer Acad. Publ., Dordrecht, 2000, 149–156 | MR | Zbl

[74] “Strongly coupled quantum discrete Liouville theory. I. Algebraic approach and duality”, Comm. Math. Phys., 219:1 (2001), 199–219 (with R. M. Kashaev, A. Yu. Volkov) | DOI | MR | Zbl

[75] “Strongly coupled quantum discrete Liouville theory. II. Geometric interpretation of the evolution operator”, J. Phys. A, 35:18 (2002), 4043–4048 (with R. M. Kashaev) | DOI | MR | Zbl

[76] “Zamechaniya o raskhodimostyakh i razmernoi transmutatsii v teorii Yanga–Millsa”, TMF, 148:1 (2006), 133–142 | DOI | MR | Zbl

[77] “Algebraic lessons from the theory of quantum integrable models”, The unity of mathematics, Progr. Math., 244, Birkhäuser Boston, Boston, MA, 2006, 305–320 | DOI | MR | Zbl

[78] “Diskretnaya seriya predstavlenii dlya modulyarnogo dublya kvantovoi gruppy $U_{q}(\operatorname{sl}(2,\mathbb{R}))$”, Funkts. analiz i ego pril., 42:4 (2008), 98–104 | DOI | MR | Zbl

[79] “Novaya zhizn polnoi integriruemosti”, UFN, 183:5 (2013), 487–495 | DOI

[80] “Moya zhizn sredi kvantovykh polei”, Vestn. RAN, 84:9 (2014), 797–804

[81] “Spektralnaya teoriya odnogo funktsionalno-raznostnogo operatora konformnoi teorii polya”, Izv. RAN. Ser. matem., 79:2 (2015), 181–204 (with L. A. Takhtadzhyanom) | DOI | MR | Zbl

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