Table of Contents
Foreword C. N. Yang v
Part 1 Scattering Theory 1
1-1 Formulas for traces for a singular Sturm-Liouville differential operator, Sov. Math. Dokl 1 (1960) 451-454 (Engl. transl.) V. S. Buslaev 3
1-2 On the Friedrichs model in the theory of perturbations of a continuous spectrum, Amer. Math. Soc. Transl., II. Ser. 82 (1967) 177-203 (Engl. transl.) 7
1-3 Increasing solutions of the Schrbdinger equation, Sov. Phys. Dokl. 10 (1966) 1033-1035 (Engl. transl.) 34
1-4 Scattering theory for a three-particle system, Sov. Phys. JETP 12 (1961) 1014 1019 (Engl. transl.) 37
Part 2 Automorphic Functions 43
2-1 Expansion in eigenfunctions of the Laplace operator on the fundamental domain of a discrete group on the Lobacevskiï plane, Trans. Moscow Math. Soc. 17 (1967) 357-386 (Engl. transl.) 45
2-2 Scattering theory and automorphic functions. J. Sov. Math. 3 (1975) 522-548 (Engl. transl.) B. S. Pavlov 75
2-3 A nonarithmetic derivation of the Selberg trace formula, J. Sov. Math. 8 (1977) 171-199 (Engl. transl.) A. B. Venkov V. L. Kalinin 102
Part 3 Field Theory 131
3-1 A remark on Sehrödinger's equation with a singular potential, Sov. Math. Dokl. 2 (1961) 372 375 (Engl. transl.) F. A. Berezin) 133
3-2 Distinction between interaction and scattering effects in perturbation theory, Sov. Phys. Dokl. 8 (1964) 881-883 (Engl. transl.) 137
3-3 An approach to the theory of the low-temperature Bose gas, Sov. Phys. JETP 20 (1965) 890- 893 (Engl. transl.) V. N. Popov)140
3-4 Asymptotic conditions and infrared divergences in quantum electrodynamics, Theor. Math. Phys. 4 (1970) 745-757 (Engl. transl.) P. P. Kulish) 144
3-5 Feynman diagrams for the Yang-Mills field, Phys. Lett. B 25 (1967) 29-30 (Engl. transl.) V. N. Popov 157
3-6 Liouville model on the lattice, in Lecture Notes in Phys., Vol. 246 (Springer. 1986), pp. 166-179 L. A. Takhtajan) 159
3-7 Hamiltonian reduction of unconstrained and constrained systems. Phys. Rev. Lett. 60 (1988) 1692-1694 R. Jackiw 173
3-8 Quantization of symplectic orbits of compact Lie groups by means of the functional integral, J. Geom, Phys. 5 (1989) 391-406 A. Alekseev S. Shatashvili 176
3-9 (T*G)t: A toy model for conformal field theory, Commun. Math. Phys. 141 (1991)413-422 A. Yu. Alekseev 192
3-10 The unravelling of the quantum group structure in the WZNW theory, preprint CERN-TH-5981/91 (.1991) A. Alekseev M. Semenov-Tian-Shansky A. Volkov 202
3-11 An involution and dynamics for the q-deformed quantum top, J. Math, Sci. 77 (1995) 3137-3145 (Engl. transl.) A. Yu. Alekseev 218
3-12 Quantum symmetry in conformal field theory by Hamiltonian methods, in New Symmetry Principles in Quantum Field Theory (NATO Adv. Sci. Inst. Ser. B Phys., Vol. 295) (Plenum, 1992), pp. 159-175 227
3-13 Abelian current algebra and the Virasoro algebra on the lattice, Phys. Lett. B 315 (1993) 311-318 A. Yu. Volkov 244
3-14 Hirota equation as an example of an integrable symplectic map, Lett. Math. Phys. 32 (1994) 125-135 A. Yu. Volkov 252
3-15 Discrete evolution for the zero modes of the quantum Liouville model, J. Phys. A 41 (2008) 194008 (Engl. transl.) A. Yu. Volkov 263
Part 4 Theory of Solitons 275
4-1 Korteweg-de Vries equation: A completely integrable Hamiltonian system, Fund. Anal. Appl. 5 (1971) 280-287 (Engl. transl.) V. E. Zakharov 277
4-2 Some comments on the many-dimensional solitons, Lett. Math. Phys. 1 (1976) 289-293 285
4-3 Quantum-mechanical approach to completely integrable field theory models, Sov. Phys. Dokl 23 (1978) 902-904 (Engl. transl.) E. K. Sklyanin 290
4-4 What is the spin of a spin wave?, Phys. Lett. A 85 (1981) 375-377 L. A. Takhtajan 293
4-5 Spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model, J. Sov. Math. 24 (1984) 241-267 (Engl. transl.) L. A. Takhtadzhyau 296
4-6 Hamiltonian structures for integrable models of field, theory, Theor. Math. Phys. 56 (1983) 847-862 (Engl. transl.) N. Yu. Reshetikhin 323
4-7 Local Hamiltonians for integrable quantum models on a lattice, Theor. Math. Phys. 57 (1984) 1059-1073 (Engl. transl.) V. O. Tarasov L. A. Takhtadzhyan 339
4-8 Instructive history of the quantum inverse scattering method, Acta Appl. Math. 39 (1995) 69 84 354
4-9 How the algebraic Bethe Ansatz works for integrable models, in Symétries Quantiques (Proc. of Les Houches, Session LXIV, 1995) (North-Holland, 1998), pp. 149-219 370
4-10 High energy QCD as a completely integrable model, Phys. Lett. B 342 (1995) 311-322 G. P. Korchemsky 440
Part 5 Quantum Groups 453
5-1 Quantum groups, in Braid Group, Knot Theory and Statistical Mechanics (World Scientific, 1989), pp. 97-110 N. Reshetikhin L. Takhtajan 455
5-2 Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990) 193-225 (Engl. transl.) N. Yu. Reshetikhin L. A. Takhtadzhyan 469
5-3 Quantum dilogarithm. Mod. Phys. Lett. A 9 (1994) 427-434 R. M. Kashaev 502
5-4 The differential calculus on quantum linear groups, in Con- temporary Mathematical Physics (AMS Transl. Ser. 2, Vol. 175) (Amer. Math. Soc., 1996), pp. 35-47 P. N. Pyatov 510
5-5 Modular double of a quantum group, Math. Phys. Stud. 21 (2000) 149-156 523
Part 6 Knots 531
6-1 Stable knot-like structures in classical field theory, Nature 387 (May 1997) 58 -61 A. J. Niemi 533
6-2 Knotted solitons, in Proceedings of the International Congress of Mathematicians 2002, Vol. I, pp. 235- 244 537
Part 7 General Questions 547
7-1 Modern mathematical physics: What it should be, in Mathematical Physics 2000 (Imperial College Press, 2000), pp. 1-8 549
List of Publications 557