1 Introuction To ene everthing precisel we consier a specic moel. Let us concentrate our attention on Bose gas with elta interaction (quantum Nonlinea

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1 The New Ientit for the Scattering Matrix of Exactl Solvable Moels 1 Vlaimir Korepin an Nikita Slavnov z ITP, SUNY at Ston Brook, NY , USA. korepininsti.phsics.sunsb.eu z Steklov Mathematical Institute, Gubkina 8, Moscow , Russia. nslavnovmi.ras.ru Abstract We iscovere a simple quaratic equation, which relates scattering phases of particles on Fermi surface. We consier one imensional Bose gas an XXZ Heisenberg spin chain. 1 Deicate to J. Zittartz on the occasion of his 60th birtha 1

2 1 Introuction To ene everthing precisel we consier a specic moel. Let us concentrate our attention on Bose gas with elta interaction (quantum Nonlinear Schroinger equation). The Hamiltonian of the moel is H = Z x x (x) x (x) + c (x) (x) (x) (x)? h (x) (x) : (1.1) Here 0 < c < 1, h > 0 are the coupling constant an the chemical potential respectivel. The canonical Bose els (x; t); (x; t); (x; t R) obe the stanar commutation relations [ (x; t); (; t)] = (x? ): (1.) an act in the Fock space with the vacuum vector j0i, which is characterize b the relation: (x; t)j0i = 0: (1.3) Alternativel the moel can be formulate on the language of man-bo quantum mechanics sstem, containing N ientical particles. In this case the Hamiltonian of the Bose gas can be represente as NX H N =? j=1 x j + c X N j>k1 (x j? x k )? hn: (1.4) For nonzero value of the coupling constant the Pauli principal is vali (chapter VII of [1]). The moel was solve b Bethe Ansatz []. The groun state is a Fermi sphere. In orer to escribe it precisel it is convenient to introuce spectral parameter (similar to rapiit). The erivative of the momentum of the particle with respect to the spectral parameter is k() = (); (1.5) where the function () is ene b an integral equation ()? 1 K(; )() = 1 : (1.6)

3 Here q is the value of the spectral parameter on the Fermi surface, an K(; ) = c c + (? ) : (1.7) One can prove that the integral operator ^I? 1 ^K is not egenerate, an hence, the equation (1.6) has unique solution ([3], chapter I of [1]). The ensit of the gas is given b D = () : (1.8) There is one particle in the moel. It is ene at q or. The energ of the particle "() is "()? 1 K(; )"() =? h: (1.9) It vanishes on the Fermi surface "(q) = 0. The momentum is Here k() = + () = i ln (? )() : (1.10) ic + ic? : (1.11) One can calculate a scattering matrix of particle with spectral parameter on another particle with spectral parameter (chapter I of [1]). There is no multi-particle prouction or reection. Transition coecient is The phase F (j) is ene b an integral equation F (j)? 1 expfif (j)g: (1.1) K(; )F (j) = 1 (? ): (1.13) The most important are scattering phases of particles on the Fermi eges F (qjq) an F (qj? q). In this paper we prove the ientit et 1? F (qjq) F (qj? q)?f (jq) 1 + F (j? q) 3 = 1: (1.14)

4 This is the main result of the paper. Another wa to rewrite this ientit is 1? F (qjq)? F (qj? q) = 1: (1.15) Here we have use the propert F (?j? ) =?F (j), which follows immeiatel from the antismmetr of (? ) =?(? ). This ientit also permit us to relate \fractional" charge to the phase shift on the Fermi surface. Fractional charge Z appears in formul for nite size corrections (chapter I of [1]). This value is necessar for conformal escription of the moel (chapter XVIII of [1]) an it is equal to Z = (q): (1.16) Using the equations (1.6), (1.13) for () an F (j), one can n the relationship between the fractional charge an the scattering phase on the Fermi surface Z = 1 + F (qj? q)? F (qjq): (1.17) Inee, it follows from (1.6) that [()?1]? 1 K(; )[()?1] = 1 Comparing this equation with (1.13) we n [(+q)?()]: (1.18) () = 1 + F (j? q)? F (jq); (1.19) what, in turns, implies (1.17). The ientit (1.15) allows us to n new relation Z?1 = 1? F (qj? q)? F (qjq): (1.0) The proof of the main ientit In this section we give the proof of the ientit (1.14). In orer to o this, one shoul calculate the erivatives of the function F (j) with respect to, an q. Using the basic equation (1.13), we have F (j)? 1 F (j) K(; ) = 1 K(; )? 1 1 K(; q)f (qj) + K(; )F (j); (.1) 4

5 F (j)? 1 F (j) K(; ) =? 1 K(; ); (.) F (j) q? 1 F (j) K(; ) q = 1 1 K(; q)f (qj) + K(; )F (j): (.3) Here we have use that (? ) = K(; ). As we have mentione alrea, the resolvent of the operator ^I? 1 ^K exists an it is equal to 1?1 1 ^R = ^I? ^K ^K: (.4) The erivatives of the function F (j) can be expresse in terms of the resolvent F (j) F (j) F (j) q = R(; )? R(; q)f (qj) + R(; )F (j) =?R(; ) = R(; q)f (qj) + R(; )F (j) (.5) Using these equations one can n the complete erivatives with respect to q of functions F (qjq), F (qj? q) etc.: F (qj q) = F (j q) =? + q F (j) =q =q + F (j) q ; = =q : (.6) 5

6 Substituting here equations (.5) we n F (qj? q) = R(q; ) 1 + F (j? q) ; F (jq) =?R(; q) 1? F (qjq) ; F (qjq) = R(q; )F (jq); (.7) F (j? q) = R(; q)f (qj? q): Now it is sucient to take the erivative with respect to q of the l.h.s. of the equation (1.14): et 1? F (qjq) F (qj? q)?f (jq) 1 + F (j? q) F (qjq) =? 1 + F (j? q) + 1? F (qjq) F (j? q) F (qj? q) F (jq) + F (jq) + F (qj? q) =?R(q; )F (jq) 1 + F (j? q) +R(; q)f (qj? q) 1? F (qjq) +R(q; )F (jq) 1 + F (j? q)?r(; q)f (qj? q) 1? F (qjq) = 0: (.8) 6

7 On the other han, it is clear that for q = 0 we have et 1? F (qjq) F (qj? q)?f (jq) 1 + F (j? q) = 1: (.9) q=0 Thus, the ientit (1.14) is prove. We woul like to emphasize the we i not use the explicit expressions for the kernel K(; ) an the function (? ). In fact, we have use onl three properties: a) the existence of the resolvent of the operator ^I? 1 ^K; b) the kernel K(; ) an the function (? ) epen on the ierence; c) the erivative of (? ) is equal to the kernel K(; ). In orer to reuce (1.14) to the ientit (1.15) one shoul use also the antismmetr propert (? ) =?(? ). Thus, the quaratic ientit for the scattering phase is vali for a wie class of completel integrable moels, but not onl for the one-imensional Bose gas. In particular, it is vali for scattering phases of elementar particles (spin waves) of XXZ Heisenberg spin chain in a magnetic el ([4], see also chapter II of [1]). Acknowlegments This work was supporte b the National Science Founation (NSF) Grant No. PHY , an the Russian Founation of Basic Research Grant No References [1] V. E. Korepin, N. M. Bogoliubov an A. G. Izergin, Quantum Inverse Scattering Metho an Correlation Functions (Cambrige Universit Press, 1993) [] E. H. Lieb an W. Liniger, Phs. Rev. 130,1605 (1963) [3] C. N. Yang an C. P. Yang, J. Math. Phs. 10,1115 (1969) [4] C. N. Yang an C. P. Yang, Phs. Rev. 150,31, (1966) 7