Vadim Kuznetsov. Informal Biography by Eyes of His First Adviser

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 3 (2007), 074, 6 pages V adim Kuznetsov. Informal Bio graph y b y Ey es of His First Advis er ⋆ Igor V. KOMAR OV St. Petersbur g State University, St. Petersbur g, R ussia E-mail: komar ov@p ob ox.spbu.ru Received May 20, 2007; Published o nline June 22, 2007 Original article is av ailable a t ht tp:// www.em is.de/journals/SIGMA/2007/074/ Abstract. The pa p er is dedica ted to the memo ry of prominent theor etical physicist and mathematician Dr. V adim Kuznetsov who w orked, in particular, in the f ields of the no nlinear dynamics, s eparation of v aria bles, integrability theory , sp ecia l functions. It includes his short resear ch biog r aphy , an ac c ount of the sta rt of his research caree r a nd the lis t of publicatio ns . Key wor ds: classical and qua nt um int egrable systems; separatio n of v ariables 2000 Mathematics Subje ct Classific ation: 01A70 V a dim Ku znetso v was a classical “self-made man”, nobo dy of his family ev er earned their living by in tel lectual w ork. V adim’s talen t and will was th e main source of his ac hiev emen ts and authorit y among the p rofessionals. The external canv as of his b iography lo oks streamlined: th e Leningrad Univ ersit y , the p ost- graduate course at the same u niv ersit y , defence of the thesis and wo rk of a researc her. In April 1993 V adim got h is f irst p ost-do ctoral p osition (Departmen t of Mathematics, Univ ersit y of Amsterdam, a NW O p ost-do ctoral researc h fello w in v olv ed in the pro ject “Sp ecial fun ctions and quan tum in v erse scattering m etho d” sup ervised by Prof. T.H. Ko ornwinder). In 1996 V. Kuznetso v sp ent six months at the T ec hnical Universit y in Lyngby , Denmark, w here he lectured in classical and quantum in tegrable systems. That was follo wed by t w o months of w ork at the Cen tre de rec herc hes math ´ ematiques, Universit ´ e d e Montr ´ eal as a researc h f ello w . In 199 6 V adim Kuznetso v f inally settled in Leeds (UK) were he w ork ed at the unive rsit y , f irst on th e gran t basis and then at a p ermanent p osition. In 1980 V adim K uznetso v entered the Departmen t of Ph ysics of the Leningrad Univ ersit y , where since 198 2 he had chosen the ma jor at the d ep artmen t of quantum mec hanics. I was a lec- turer then at the same dep artment, and I had to giv e a course w ith a strange title “In tro duction to Profession”. T he student s had not studied quantum mec hanics at that moment, and it was not clear what sh ould b e taugh t. The administration told me to do wh at I wished to. I ha v e c hosen a system of talks based on b o ok th en recen tly pu blished in Russian by W. Miller “Sepa- ration of V ariables”. Some of the stud en ts w ere giv en pap ers in other topics. V adim got a pap er from J ournal of the O ptic al So ciety of Americ a on halo ef f ect at atmosph eric light scattering on ice crystals. With r esp ect to the talk, V adim s ho w ed p ersistence in studying of mathematical and p h ysical d etails of the pap er. Then I got an imp ression of him as an accurate p erson who is go o d in lab orious tasks. Later I asked V adim whether this cour s e had any inf luence up on him. The answ er w as negativ e. Ma yb e b ecause of this course V adim d id not feel any “trepidation” with resp ect to the classical theory of separation of v ariables. In around a y ear V adim o v ertook me in a stairw a y an d ask ed for a topic for r esearc h. On m y wa y I suggested him to f in d a s emi-classical sp ectrum of the Ko w ale vski top. I had tw o ideas – to use K oloso v’s pro jection of the Ko w alevski problem onto a f lat p otent ial p roblem and ⋆ This pap er is a con tribution to t he V adim Kuzn etso v Memorial Issue ‘Integrable Systems and Related T opics’. The full collection is av ailable at http://www. emis.de/journals/SIGMA/kuznetso v.html 2 I.V. Komaro v to apply the adiabatic switc hing metho d for f ind ing of the sp ectrum. I b eliev ed then that it was a sure-win pr oblem needing only lab or in put. W e started regular and intense discuss ions on this topic. In a few months V adim brought me inte grals of action calculated in line with the K oloso v’s metho d. I asked him to in v estig ate limiting cases for wea k and strong f ields. In one of the limits one degree of fr eedom was disapp earing. The formulas w ere cum b ersome, and I b eliev ed that to b e a r esult of some tec hnical mistak e. W e got stuc k at that place for sev eral months. Gradually it b ecame clear that the reason for the problem is th at the K oloso v ’s transformation where time also transforms is not canonical. V adim Kuznetso v found an indicatio n how to pro ceed correct ly in the pap er by S .P . No vik o v a nd A.P . V eselo v: f ir st of all, Kow al evski v ariables should hav e b een transformed into the Poisson comm uting ones. W e inv ented the next step ourselve s. I t w as restoration of the canonical v ariables when the integ rals of motion in the Lagrangian v ariables w ere kno wn. F or me and probab ly f or V adim that w as a v ery happy moment. A student in his v ery f irst w ork started to generate professional ideas at the serious level and got assured in his researc h capacities. Appro ximately at the same time E.K. Sklya nin includ ed the Gory ac hev–Chaplygin gyrostat I ha v e stu died into the framework of the qu an tum inv erse scattering m etho d (QISM). The Skly anin’s metho d had v ario us names – “ the fun ctional Bethe ansatz” , “the m agical recip e”. It w as ob vious that it dealt with f inding of the separation of v ariables (S oV) in the QISM. I b rought V.B. Kuznetso v and his classmate A.V. Tsiga no v to L.D. F adeev’s seminar at the Leningrad Branc h of Steklo v Mathematic al Ins titute an d introdu ced th em to E .K . S kly anin. V. Kuznetso v thoroughly stu died eac h n ew p ap er b y E.K. S k lyanin. In particular, s ome time later V adim studied the algebra of the ref lection equ ations (we called it then Q ISM I I), and he w as the f irst to constr u ct the separation for one example of QIS M I I. A t that ti me V adim needed to complete te c hnically h is education and to defend a master thesis on the basis of the results already obtained. It b ecame clear to us that deve lopmen t and application of the separation of v ariables in th e QIS M is a priority task. I t wa s muc h later when in the pap er [43] “Ko w alevski top revisited” in 2002 V adim got bac k to th e top problem and found explicit and quite cumbersome expressions for 2 × 2 of the Lax matrix for the Ko w alevski top. The results of his master thesis w ere publish ed in [1] in The or etic al and Mathematic al Physics in 1987. In 1986–198 9 V adim w as a p ost-graduate stud en t at the department of computational physics to wh ic h I mo v ed some time b efore. Our meetings were regular and f ruitful, V adim w as quickly b ecoming a qualif ied and ind ep endent researc her. I set some problems for him, and V adim found some p roblems himself. W e also discussed strategic issues – in w h at d irection it is preferable to con tin ue researc h in future. V adim resp onded p ositiv ely to extension of the list of the integ rable systems, incorp oration of the kn own in tegrable systems int o the QIS M sc hemes, accum ulatio n of the v ariable separation exp erience in p articular cases. On the opp osite, V adim rejected m y suggestion to lo ok inside the QI S M f or ef f icien t algorithms for f in ding the sp ectrum of the in - tegrals of m otion that were naturally related to the QIS M. Among the “consumers” of the new ideas of the SoV metho d , the one mentio ned most often wa s the theory of sp ecial functions that w e p erceive d in th e s p irit of the th r ee-v olume b o ok by Bateman and Erdely . Sev eral Lax-lik e repr esen tations for the Ko w alevski pr oblem app eared in the 1980-ies. V adim rewrote the a v ailable direct deriv atio n of the separated equations for the case so (4 , C ) in t he spirit of Heine–Horozo v. A t the same time A.G. Reiman and M.A. Semenov-T yan-Shanskii constructed 4 × 4 and 5 × 5 Lax matrices for th e original K o w alevski problem. T he new Lax matrices depr eciated the previous v arian ts and allo wed simplif ication of the classical equations of motion in terms of Pr ym θ -fu nctions. T h e attempts to f ind separation of v aria bles from the new Lax matrices d id not lead to an y su ccess. As V adim p ointed out later [43] (Kow alevski top revisited, 2002), until then there w as only one separation of v ariables kno wn for this pr ob lem follo wing from original Ko w alevski’s pap er s . V adim had identif ied in a 4 × 4 minor of the V adim Kuznetso v. Informal Biograph y by Eye s of His Firs t Adviser 3 Lax matrix f or the Ko w alevski pr oblem a new 3 × 3 Lax matrix for the Gory ac h ev–Chaplygin gyrostat. With A.I. Bob enk o they had applied this matrix for in tegration of the resp ectiv e equations of motion [3 ]. After the p ost-graduate course and defence of his thesis (referees E.K. Skly anin and A.R. Its) earlier than the sc heduled date in Octob er of 1989 V. Kuznetsov w ork ed as a researcher at the Departmen t of Compu tational Physics. In autumn of 1991 Ern ie Kalnins that wo rk ed f or many y ears on the coord inate s eparation of v ariables visited our univ ersit y . Ern ie w an ted to see Riga, and V adim accompanied h im on that trip. Before E rnie Kalnins’s visit V adim wrote on his o wn initiativ e sev eral pap er s on co-ordinate separation of v ariables for free motion in the sp aces of p erman ent curv ature in classical and qu an tum mec hanics on the basis of currents algebra [9 , 10, 11]. The most imp ortant wa s that it was p ossible not o nly to r ep ro duce quite transparen tly the kno wn re- sults on c o-ordinate separation of v ariables for the Laplace op erators in th e lo w-dimen s ional spaces of constan t curv ature from the Gaudin alge bra, but also to extend this pro cedure to the sp aces with arbitrary num b er of dimens ions and to construct reduction of quadrics. These results attracted the considerable interest at the conference in Obnin sk of N.Y a. Vilenkin wh o ask ed questions to V adim for quite a long time and in great detail. Later (F eb ruary 25, 1993) V adim r eceiv ed f or this cycle of pap ers the Award of the Academiae Eu ropaeae (London), instituted by the club of the Russian m emb ers of the Academiae for young Russian scien- tists. The end of the 1980-ies and the b eginnin g of th e 1990-ies were a dif f icult time for sur viv al. The USS R econom y w as collapsing fast. The researc her’s salary was evident ly insuf f icien t to supp ort th e family . V adim found o dd jobs, u p to p ett y trade near metro stations th at was v ery humiliating for him. V adim f ou n d a p ostdo ctoral p osition at the Univ ersit y of Amsterdam. I learned ab out h is decision to go abroad when V adim asked me for a letter of recommendation. In Amsterdam V adim work ed together with the exp ert on the sp ecial fu nctions theory T h om H. Ko ornwinder. The results of their joint researc h w ere included into the 1994 pap ers [18, 25] w ere prop erties of the Gauss hyp er geometric function (in p articular, con tiguous fu nction relations) w ere deriv ed in a regular wa y from the prop erties of the R -matrix algebra. The pap ers of this p erio d develo p ed the ideas of the separation of v ariables for rational and tr igonometric linear and quadratic R -matrix algebra. The list of V adim’s co-authors wa s extending fast. V adim adv anced his knowledge and mastered new f ields of mathematics and theoretical p h ysics. In 1994 a man y-y ear co-op eration of V.B. K uznetso v and E.K. S kly anin h ad started. They obtained separation of v ariables for th e A 2 Jac k p olynomials. Systematic researc h in the f ield of the Ruijsenaars–Sc hneider mo del w as also started [26] (the co- authors w ere F.W. Nijhof f, O. Ragnisco, E.K. Sklya nin). I w ould like to mention here the observ ati on on the r elation of the Bethe equations w ith the integrable time-discretisatio n of the mo del. It is k n o wn that for the separation of v ariables b y means of the p oles of the Bak er–Akhieze r function f its sp ecial normalisation is needed. F or m an y examples the scalar pro d uct ( f , a ) with a constant v ector a w as suf f icien t, b ut sometimes the n umber of p oles exceeded th e n um b er of degrees of freedom. In the pap er on D n t yp e p erio dic T od a lattice V. Ku znetso v managed to f ind a correct n orm alising vecto r a dep endin g on the dynamic v ariables [30]. But that was his last work in v olving p oles of the Bak er–Akhiezer fu nction. Presence of artif icial tric ks made V adim to lo ok for more univ ersal app roac hes. The p ap ers b y V.B. Kuznetso v and E.K. Skly anin [32] dev elop ed a new more systematic ap - proac h to s ep aration of v ariables and p resen ted relation of the SoV with th e B¨ aklund transfor- mation. Inv en tion of the “sp ectralit y ” p rop erty w as a k ey p oin t. Th e Baxter Q -op erator b ecame a sub ject of attent ion of many r esearc hers. V ad im often said that a new un derstanding of the separation of v ariables started f or him fr om the 1998 pap er [32] “On B¨ ac klun d transformations 4 I.V. Komaro v for man y-b o dy systems”. Dev elopmen t of this tec hnique by the group of m athematicians with participation of V. Kuznetsov led to qualitativ ely new r esults on factorisation of the symmetric Jac k, Hall–Littlew o o d and Macdonald p olynomials. F or more than 10 y ears V adim maint ained researc h con tact s with th e group of Mark Adler, Pierre v an Mo erb ek e and P ol V anh aec k e, with wh om he p u blished only tw o pap ers [40, 41] on geometric asp ects of the B¨ ac k lu nd transform. In 2000 V adim organised joint ly with F rank Nijfof f th e Inte rnational W orksh op on Mathe- matical Metho d s of Regular Dynamics dedicated to the 150th ann iv ersary of Sony a Ko w alevski. In 2002 V adim K uznetso v and F rank Nijhof f edited the materials of the W orksh op [42]. In 2003 V adim was an organiser of the workshop in Edinburgh that was dedicated to the classical w orks b y H. Jac k, P . Hall, D.E. Littlew oo d and I .G. MacDonald on S ymmetric F unctions and their relation to Repr esen tation T heory of S ymmetric Group s . Th e materials of this wo rkshop edited b y V. Ku znetso v and S. Sahi [51] were published after his death. In that issue Brian D. Sleeman and Evgen y K. Sklyanin pu blished a pap er ded icated to V adim Kuznetso v’s memory 1 . V adim wo rk ed a lot directly with form ulas. Somet imes h e obtained results lik e “Hidd en symmetry of the quantum Calogero–Moser system”, 199 6, that at the f irst glance had no relation to the general theory . This list can b e extended and it may give start to fu ture studies. I visited V adim and h is family in Amsterdam, Cop enhagen and Leeds, and we also met durin g V adim’s sh ort visits to St. P etersburgh and Dubna. In Ju ly 1998 b oth of us were in d if ferent regions of German y and sp ecially came to Kaizerslautern to see eac h other. W e w ere w alking in the cit y where some Sunda y German F est w as bustling, and V adim enth u siastically spoke ab out new id eas in the join t p ublications w ith E.K. Sklya nin. Later the researc h asp ect of our discussions mo v ed to a backg round. D uring our last mee tings in Leeds in Ma y 2005 V adim lo ok ed very tired, he has immense teac hing a nd a dministrativ e load. W e tried to talk about mathematics without big su ccess. V adim mentio ned that h e in tended to start a completely new researc h area n ot related to separation of v ariables. I know nothing ab out imp lemen tatio n of these plans. V adim w as survived by his wife Olga and son Simon. Dr. V adim B. Kuzn etso v. List of publications [1] Komaro v I.V., Kuznetsov V.B., Qu asiclass ical qu antiza tion of the Kow alevski top, T e or et. M at. Fiz. 73 (1987), 335–347 (English transl.: The or. and M ath. Phys. 73 (1987), 1255–12 63). [2] Komaro v I.V., Kuznetso v V.B., The generalized Goryac hev–Chaplygin gyrostat in quantum mec hanics, in Dif ferentsialna y a Geom. Gruppy Li i Mekh ., V ol . I X, Zap. Nauchn. Sem. L eningr ad. Ot del. Mat. Inst. Steklov. (LOMI ) 164 (1987), 133–141 , 198 (English transl.: J. Soviet Math. 47 (1989), no. 2, 2459–2466). [3] Bob enko A.I., Kuznetsov V.B., Lax representa tion and new form ulae for the Gory ac hev–Chaplygin top, J. Phys. A: Math. Gen. 21 (1988), 1999–2006. [4] Kuznetsov V.B., Tsigano v A.V., A special case of Neumann’s system and the Ko w alevski–Chaplygin– Gory achev top, J. Phys. A: Math. Gen. 22 (1989), L73–L79. [5] Kuznetsov V.B., Tsigano v A .V., I nf inite series of Lie algebras and b oundary cond itions for integrable sys- tems, in Dif ferentsialna y a Geom. Gruppy Li i Mekh., V ol. 10, Zapiski Nauchn. Se m. LOMI 172 (1992), 88–98, 170 (English transl.: J. Soviet Math. 59 ( 1992), n o. 5, 1085– 1092). [6] Komaro v I.V., Kuznetsov V.B., K o w alevski’s top on the Lie algebras o(4), e(3), and o(3,1), J. Phys. A: Math. Gen. 23 (1990), 841–846. [7] Kuznetsov V.B., Generalize d polyspheroidal perio dic functions and the quantum inv erse scattering method , J. Math. Phys. 31 ( 1990), 1167–1174. [8] Komaro v I .V., Kuzn et sov V.B., Quantum Euler–Manak o v top on the 3-sph ere S 3 , J. Phys. A: Math. Gen. 24 (1991), L737–L742 . 1 Sleeman B.D., Sk lyanin E.K., V adim Boriso vic h Kuznetsov 1963–2005, Contemp. Math. 417 (2006), 357–360 . V adim Kuznetso v. Informal Biograph y by Eye s of His Firs t Adviser 5 [9] Kuznetsov V.B., On t h e equiv alence of tw o families of quan tum integrable systems, in V oprosy Kv an t. T eor. P olya Statist. Fiz., V ol. 11, Zap. Nauchn. Sem. S.-Peterbur g. Otdel. Mat. Inst. Steklov. (POMI) V ol. 199 (1992), 114–131 , 186 (English transl.: J. Math. Sci . 77 (1995), no. 2, 3090–3101 ). [10] Kuznetsov V.B., Quadrics on Riemannian spaces of constant cu rv ature. Separation of vari ables and a con- nection with the Gaudin magnet, T e or et. M at. Fiz. 91 (1992), no. 1, 83–111 (English t ransl.: T he or et. and Math. Phys. 91 (1992), no. 1, 385–404 ). [11] Kuznetsov V.B., Quadrics on real Riemannian spaces of constant curv ature: separation of v ariables and connection with Gaudin magnet, J. Math. Phys. 33 (1992), 3240–3254. [12] Kuznetsov V.B., O n isomorphism of n -dimensional Neumann system and n -site Gaudin magnet, F unktsional. Ana l. i Prilozhen. 26 (1992), no. 4, 88–90 ((English transl.: F unct. Anal. Appl. 26 (1992), no. 4, 302–304). [13] Kuznetsov V .B., Equiv al ence of tw o graphical calculi, J. Phys. A: Math. Gen. 25 (1992), 6005–6026. [14] Enol’skii V .Z., K u znetsov V .B., Salerno M., On the quantum inverse scattering method for the DST d imer, Phys. D 68 (1993), 138–152. [15] Eilbeck J.C., Enol’skii V.Z., Kuzn et sov V.B., Leyk in D.V., Linear r -matrix algebra for systems separable in parab olic coordinates, Phys. L et t. A 180 (1993), 208–214. [16] Christiansen P .L., Jørgensen M.F., Kuznetsov V.B., On integra ble systems close to the T o da lattice, L ett. Math. Phys. 29 (1993), 165–173. [17] Kuznetsov V.B.; Tsyganov A.V., Quantum relativistic T o da lattices , in D if ferentsial nay a Geom. Gruppy Li i Mekh ., V ol. 13, Zap. Nauchn. Sem. S.-Peterbur g. Otdel. Mat. Inst. Steklov. (POMI) 205 (1993), 71–84, 180 (English transl.: J. M ath. Sci. 80 (1996), n o. 3, 1802–1 810). [18] Koornwinder T.H., Kuznetsov V .B., Gauss hyp ergeometric function and quadratic R -matrix algebras, Alge- br a i Analiz 6 (1994), 161–184 (English transl.: St. Petersb ur g M ath. J. 6 (1995), 595–618), hep-th/9311152. [19] Kalnins E.G., Kuznetsov V.B., Miller W.Jr., Quadrics on complex Riemann ian spaces of constant curv ature, separation of v ariables and the Gaud in magnet, J. Math. Phys. 35 (1994), 1710–17 31, hep- th/9308109. [20] Eilbeck J.C., Enol’skii V.Z., Kuznetso v V.B., Tsiganov A.V., Linear r -matrix algebra for classica l separable systems, J. Phys. A: Math. Gen. 27 (1994), 567–578, hep-th/9306155. [21] Eilbeck J.C., Enol’skii V .Z., Kuznetsov V.B., Leykin D.V., Classical Pois son structure for a hierarch y of one-dimensional particle systems separable in parabolic co ordinates, J. Nonline ar Math. Phys. 1 ( 1994), 275–294 . [22] Kalnins E.G., Kuzn etsov V.B., Miller W.Jr., S eparation of v ariables and X X Z Gaudin magnet, R end. Sem. Mat. Univ. Poli te c. T orino 53 ( 1995), 109–12 0, hep-t h /9412190. [23] Kuznetsov V.B., Jørgensen M.F., Christiansen P .L., New b ou n dary conditions for integrable lattices, J. Phys. A: Math. Gen. 28 (1995), 4639–4654, hep -th/9503168. [24] Kuznetsov V.B., Sklyanin E.K., Separation of v ariables in A 2 type Jack p olynomials, RIM S Kokyur oku 919 (1995), 27–43, solv-int/95 08002. [25] Kuznetsov V .B., 3 F 2 (1) hypergeometric function and quadratic R -matrix algebra, in Sy mmetries and Inte- grabilit y of dif ference equations (Esterel, PQ, 1994), CRM Pr o c. L e ctur e Notes , V ol. 9, Am er. Math. So c., Pro vidence, RI, 1996, 185–197, h ep-th/9410146. [26] Nijhof f F.W., R agnisco O ., Kuznetsov V.B., I ntegra ble time-discretization of the Ru ijsenaars–Schneider mod el, Comm. M ath. Phys. 176 (1996), 681–70 0, hep-t h/9412170. [27] Kuznetsov V .B., Hidden symmetry of th e quantum Caloge ro–Moser system, Phys. L ett. A 218 (1996), 212–222 , solv-int/9509 001 . [28] Kuznetsov V .B., Sk lyanin E.K., Separation of v ariables for the A 2 Ruijsenaars system and a new inte- gral rep resen tation for the A 2 Macdonald p olynomials, J. Phys. A: Math. Gen. 29 (1996), 2779–2 804, q-alg/960202 3 . [29] Nijhof f F.W., Kuznetsov V.B., Sklyanin E.K., Ragnisco O., Dyn amical r -matrix for t he elliptic R uijsenaars– Schneider system, J. Phys. A: Math. Gen. 29 (1996), L333–L340, solv-int/9603 006 . [30] Kuznetsov V.B., Separation of va riables for t he D n type p erio dic T o d a lattice, J. Phys. A: Math. Gen. 30 (1997), 2127–21 38, solv-int/97010 09 . [31] Kuznetsov V.B., N ijhof f F.W., S klyanin E.K., Sep aration of va riables for the Ru ijsenaars system, Comm. Math. Phys. 189 (1997), 855–877 , solv-int/9701 004 . [32] Kuznetsov V.B., Sk lyanin E.K., On B¨ ac klund transformations for many-bo dy systems, J. Phys. A: Math. Gen. 31 (1998), 2241–22 51, solv-int/97 11010. 6 I.V. Komaro v [33] Kuznetsov V.B., Sk lyanin E.K., Sep aration of v ariables and integral relations for sp ecial functions, R amanu- jan J. 3 1999 , no. 1, 5–35, q-alg/9705006. [34] Kuznetsov V.B., Sklyanin E.K., F acto risation of Macdonald p olynomials, in Symmetries and Integrabilit y of Dif ference Equations (Can terbury , 1996 ), L ondon Math. So c. L e ctur e Note Ser. , V ol. 255, Cam bridge Un iv . Press, Cam bridge, 1999, 370–384, q- alg/9703013 . [35] Hone A.N.W., Ku znetsov V.B., Ragnisco O., B¨ a cklund transformations for many-b o dy systems related to KdV, J. Phys. A: Math. Gen. 32 (1999), L299–L306, solv-int/990400 3 . [36] Hone A.N.W., Ku znetsov V.B., Ragnisco O., B¨ ac klund transformations for t h e Hen on –H eiles and Garnier systems, in SIDE I I I — Symmetries and I ntegrabil ity of Dif ference Eq uations (Sabaudia, 1998), CRM Pr o c. L e ctur e Notes , V ol. 25, Amer. Math. So c., Providence, RI, 2000, 231–235 . [37] Kuznetsov V.B., Salerno M., Sk lyanin E.K., Quantum B¨ ac klund t ransformation for the integra ble DST mod el, J. Phys. A : Math. Gen. 33 (2000), no. 1, 171–189, solv-int/99080 02 . [38] Kuznetsov V.B., Nijhof f F.W. (Editors), Kow alevski wo rkshop on mathematical metho ds of regular d ynam- ics, Papers from th e w orkshop held at the Un iversi ty of Leeds (Leeds, A pril 12–15, 2000), J. Phys. A: Math. Gen. 34 (2001), no. 11. [39] Hone A.N.W., Ku znetsov V.B., Ragnisco O., B¨ ac klund transformatio ns for the sl(2) Gaudin magnet, J. Phys. A: Math. Gen. 34 (2001), 2477–2490, nlin.SI/0007041. [40] Kuznetsov V.B., V anh aecke P ., B¨ a cklund transformations for f inite-dimensional integrable systems: a geo- metric approac h, J. Ge om. Phys. 44 (2002), 1–40, nlin.SI/0004003 . [41] Adler M., Kuznet sov V.B., van Moerb eke P ., Rational solutions to the Pfaf f lattice and Jack p olynomials, Er go dic The ory Dynam. Systems 22 (2002), 1365–1405, nlin.SI /020203 7 . [42] Kuznetsov V.B. (Editors), The Kow alevski prop erty , Pro ceedings of the Kow al evski W orkshop on Mathe- matical Meth od s of Regular Dyn amics (MMRD), dedicated to t h e 150th anniversary of Sophie Kow alevski’s birth ( U niversit y of Leeds, April 12–15, 2000), CRM Pr o c e e dings and L e ctur e Notes , V ol. 32, Amer. Math. Soc., Providence, R I, 2002. [43] Kuznetsov V.B., Kow alevski top revisited, in The Kow alevski Prop erty (Leeds, 2000), C RM Pr o c. L e ctur e Notes , V ol. 32, Amer. Math. So c., Pro vidence, RI, 2002, 181–196, n lin.SI/0110012. [44] Kuznetsov V.B., Simultaneous separation for the Kow alevski an d Gory ac hev–Chaplygin gyrostats, J. Phys. A: Math. Gen. 35 (2002), 6419–6430, nlin.SI /020100 4 . [45] Kuznetsov V.B., Inverse problem for SL(2) lattices, in SPT 2002: Symmetry and P erturbation Theory (Cala Gonone), W orld Sci. Publishing, River Edge, NJ, 2002, 136–15 2, nlin.SI/0207025. [46] Kuznetsov V.B., Mangazeev V .V., Sklyanin E.K., Q -op erator and factori sed separation chain for Jack p olynomials, Indag. Math. (N. S.) 4 (2003), 451–482, math.CA/0306242. [47] Kuznetsov V.B., Petrera M.,, R agnisco O., Separation of v ariables and B¨ ac klund transformations for the symmetric Lagrange top, J. Phys. A: Math. Gen. 37 (2004), 8495–85 12, nlin.SI/0403028. [48] Y uzbashy an E.A., Altshuler B.L., Kuznetsov V.B., En olskii V .Z., Solution for the dynamics of the BCS and central spin p roblems, J. Phys. A: Math. Gen. 38 (2005), 7831–7849, cond-m at/0407501 . [49] Y uzbashy an E.A., Altshuler B.L., Kuzn etso v V.B., Enolskii V .Z., Nonequilibrium Coop er pairing in the nonadiabatic regime, Phys. R ev. B 72 (2005), 220503(R), 4 pages, cond- mat/0505493. [50] Y uzbashy an E.A., Kuznetsov V .B., A ltshuler B.L., Integrable dyn amics of coupled F ermi–Bose condensates, Phys. R ev. B 72 (2005), 144524, 9 pages, cond-mat/0506782. [51] Kuznetsov V.B., Sahi S. (Editors), Jack, Hall–Littlewood and Macdonald p olynomials, Pro ceedings of th e W orkshop held in Edinburgh ( S eptember 23–26, 2003), Contemp. Math. 417 (2006). [52] Kuznetsov V.B., Sahi S., Preface, in Jac k, Hall–Littlew oo d and Macdonald P olynomials, Editors V.B. Kuznetsov and S. S ah i, Contemp. Math. 417 (2006), ix –x vii. [53] Kuznetsov V.B., Sk lyanin E.K., F actorization of symmetric p olynomials, in Jack, Hall–Littlew ood and Macdonald Polynomials, Editors V.B. Ku znetsov and S. Sahi, Contemp. Math. 417 (2006), 239–2 56, math.CA/050125 7 . [54] Kuznetsov V.B., Sklyanin E.K., Eigenproblem for Jacobi matrices: hyp ergeometric series solution, math.CO/05092 98 . [55] Kuznetsov V.B., Orthogonal p olynomials and separation of v aria bles, in Orthogonal P olynomials and Sp ecial F unctions, L e ctur e Notes in Math. , V ol. 1883, S p ringer, Berlin, 2006 , 229–254 .