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Bethe Ansatz and the generalized Yang-Baxter equations
ANNALS
OF PHYSICS
176, 2248
(1987)
Bethe Ansatz and the Generalized Yang-Baxter Equations EUGENE GUTKIN* Department
of Marhemarics, Los Angeles,
Received
September
University California
20, 1985; revised
of Southern 90089 October
California.
17, 1986
INTRODUCTION
Bethe Ansatz plays an important role in the mathematical physics (see [S] and review [16]). Starting with the work of Bethe [3], Bethe Ansatz became an indispensable tool for solving various models of mathematical physics. It suffices to mention the classical papers [Z, 13, 14, IS] on Bethe Ansatz for the delta Bose gas, Baxter’s papers on Bethe Ansatz in statistical mechanics (see [ I] and references therein), the work of Leningrad’s school on Bethe Ansatz in quantum field theory (see, e.g., [ 173 and references therein). Obviously, the wide applicability of Bethe Ansatz is due to the fact that Bethe Ansatz in different fields means different things which are connected with each other in many ways. In this article we investigate the Bethe Ansats in the nondiffractive (nonrelativistic) quantum many-particle scattering. This framework goes back to the paper [2] on the scattering of particles with delta interaction. Our point of view is as follows. Suppose we have a system of N particles in one dimension which interact only at the moments of collisions. Thus the wave function of the system on the space R” = {x, ,..., xN} should be the same as the wave function of the free (noninteracting) system outside the union of “hyperplanes of collision” {-xi = x,, 1 < i < j 6 N}. These hyperplanes divide RN into N! disconnected chambers C,,. labeled by the elements w of the permutation group W,. In each C,. the wave function f of an eigenstate of the system should coincide with in eigenstate of the free system, i.e., a linear combination of plane waves. These special many-particle eigenstates are the Bethe Ansatz wave functions f: Transition rules from a chamber to another chamber should be the same for all Bethe Ansatz eigenstates J They are given by the matching operators (coefficients ,4(n, p), B(;1, p) (see Fig. 2) which are determined by the two-particle interaction (Hamiltonian) of the problem. Thus the physics of the problem determines these matching operators X4(,I, p), B(1, p), which is, essentially, the two-body scattering. Now the mathematical * Partially
supported
by NSF
Grant
DMS
86-00350.
22 0003-4916/87
$7.50
Copyright CI 1987 by Academic Press, Inc. All rtghts of reproduction in any form reserved.
BETHE ANSATZ AND YANG-BAXTER
EQUATIONS
23
properties of the matching operators shouid determine whether the Bethe Ansatz holds for the N-particle eigenstate for all N> 2. This determination is the subject of the present work. Our framework is sufficiently general which allows us to consider the scattering of particles of many kinds and of many external energy levels. We denote the number of kinds of particles by p and the number of external energy levels by d. We also consider the scattering of particles of arbitrary statistics (the full Bethe Ansatz). Our starting point is the observation that the full Bethe Ansatz holds if and only if the coeflicients A(L p), @I., II) satisfy a special system of functional equations. The general case reduces to the two special cases: (i) (ii)
p= 1, r/a I and p> 1. d= 1.
The case (i) is much simpler. We find the general solution of the system of functional equations (Theorem I), i.e., the general expression for .4(/., p), B(E..,I(). which satisfy the full Bethe Ansatz. Then we investigate the Bethe Ansatz for the Fermi and Bose statistics. It turns out that for the Fermi statistics the Bethe Ansatz is trivially satisfied. The Bose case is more interesting. We show that the Bethe Ansatz is satisfied if the matching operators commute with each other for all values iv, p of the wave numbers and find explicit formulas for the Bethe Ansatz eigenstates (Theorems 2-4). Thus for the many-particle scattering of type (i) we solve the Bethe Ansatz problem completely. In Section 5 we write the system of functional equations on the matching coefficients which is equivalent to the Bethe Ansatz in case (ii), i.e., the scattering of many kinds of particles. This system of six equations contains, in particular, the famous Yang-Baxter equation. The Yang-Baxter equation comes up in the study of the factorizable S-matrices [ 191 and the soluble lattice models of statistical mechanics [I]. Many solutions of the Yang-Baxter equation are found in the literature (cf. [ 121). In view of the above we call our system of six functional equations the generalized Yang-Baxter equations. We postpone a detailed study of these equations to another publication. In Section 6 we illustrate the results by a few examples.
1. TWO-PARTICLE
SCATTERING
A particle is characterized by its position s, its kind i and its external energy level k. Assuming that we have p kinds of particles and d external energy levels we consider the linear space CP generated by the basic vectors r,, 1 B id p, and the space C” generated by sk, 1 ,< k
expressions f(.)c)e Q s, i.e., the space of such states is the tensor L,(R)QCPQC?
product
24
EUGENE
GUTKIN
Since the general energy levels are the same for all particles, the two-particle states f(x, y) ei@ ej@ sk live in the space L,(R2) @ (Cp @ CP) 0 C4 The particles with positions at x and y do not interact outside the collision line {x = y}. Thus we look for the eigenstates of the two-particle Hamiltonian in the Bethe Ansatz form: exp(fi (Lx + py)u 0 u, where I and p are the wave numbers, u E CP 0 Cp, u E Cd. The interaction line {x = y} divides the xy-space into the two noninteracting regions {x > y } and {x < y }. A Bethe Ansatz eigenstate f with wave numbers L and p given by .f=exp(fi(Ix+py))uOr
(1.1)
in the region {x < y}, transforms into f = exp(fi
(2-x + w)) &A P)(U 0 u)
+ exp(fi
(CLX+ 2~)) &A P)(U 0 u)
(1.2)
in the other region {x > y}. The operator-valued functions A”(2, p) and &,I, p) are called the matching operators for the two-particle scattering. Note that A”(L, p) and &A, p) operate on the space CP 0 Cp 0 Cd. They are determined by the two-particle Hamiltonian or, equivalently, by the boundary conditions for the eigenstates. As a result of collision of two particles i and j which were in the state k we obtain particles i’ and j’ in the state k’. The transition k + k’ does not depend on the kinds of particles, therefore the matching operators must be of the form A”(4 11)= T(A PL)0 AtA 111, m
P) = RCA 11)0 B(k
PI,
(1.3)
where T, R operate on Cp 0 Cp and A, B operate on Cd. Continuity of the eigenstates f across the collision line implies A”(il, p) + B(A, p) = 1.
(1.4)
If the matching operators are consistent, matching backwords, from (x > y} into {x < y } we must recover the original eigenstate (1.1). The consistency conditions are B2(A, p) + A”@, 1) ‘qn, p) = 1
(1.5)
A(& p) B(ll, p) + B(p, 1) &A, ,u) = 0.
(1.6)
and
By (1.4) A(,?, p) and B(1, p) commute and, in view of (1.3), T(A, p) commutes with R(A, p) and A(1, p) commutes with B(1, p). Assume that d> 1 and let uk,
BETHE ANSATZ AND YANG-BAXTER
EQUATIONS
35 -_
1 6 k d d, be the common eigenvectors of A and B with the eigenvalues Us and h, Applying (1.4) to the vector u@ ilk and using (1.3), we obtain (a,T+h,R)l(O~,=uOo,,
(1.7)
which implies a,TSh,R=
1
(1.X)
for all h-. Assume that not all eigenvalues ah are equal or that not all h, are equal. for concreteness, (a,, h, ) # (a?, h,). Then (a,-a,)T+(h,
-h,)R=O,
(l.(J)
which, together with (1.8), implies that T and R are scalar operators. Denoting these scalars by t( A, p) and r( I., p) we have from ( 1.3) (1.10) That is, up to a normalization, T= R = 1, there is no interaction between different kinds of particles and the particles of each kind scatter on each other by the same rules. Thus we can assumewithout loss of generality that there is only one kind of particle. Analogously, assuming that p > 1 and that the matching operators T, R are not scalar, we come to the conclusion that there is no scattering between various energy levels. thus we can assumewithout loss of generality that d= 1. To summarize, the general many-particle scattering problem set up above reduces to the following two cases. (i) Particles of one kind, p = 1. the scattering is between various external energy levels. (ii) Particles of many kinds in the absence of the external field, d= 1. The scattering is between the different kinds of particles. In this article we study in detail the case (i) which is the scattering of particles of one kind. For brevity, we call it the scattering of type one. Case (ii) which is the scattering between different kinds of particles is considered in Section 5.
2. BETHE ANSATZ
Let N 3 2 and let W, be the permutation group of [ l,..., Nj. Let E, be a finitedimensional vector space over C with an action P + 11~of IV,. The group U’, naturally acts on RN by x -+ WY and on the space C(R,‘, EY) of continuous functions on RN with values in E, by (yf)(s)
= wf’(w ‘.Y).
(2.1)
26
EUGENE
GUTKIN
We denote elements of RN by x=(x ,,..., x,,,), y = (yl ,..., yN) and let (x, y) = CyI, x,y, be the scalar product. For 1= (2, ,..., ~,,,)EP denote by e”(x) the exponential function exp(fi (A, x ) ) and call elements of C(RN, EN) of the form f(x) = ue’(x), u E E, the plane waves. We use notation
f =1
Uj.e’:(X)
(2.2)
for finite linear combinations of plane waves. For 1 < i < j d N denote by h, the hyperplane {xi = xi} in RN, by sij E W the reflection xi + Xi, xi -+ xi of RN about h, and set RN, = {x1 < . . . < x,}. The following facts are well known. Reflections si,;+ ,, i= l,..., N- 1, generate the group W and RN is the union of “chambers” C,,. = HJRN,, w E W. The rule w -+ C,,. establishes a one-to-one correspondence between chambers and elements of W. Two chambers CL,= URN, and C,,. = WRY are called adjacent if they have a common “wall” hi,. A sequence C I ,..., C,, of chambers is called a gallery if for any k = l,..., n - 1 the chambers C, and C,,, are adjacent. A gallery starting of the “fundamental chamber” C, = RN, and ending at C,,. defines a decomposition w=sj
(2.3)
n j”“‘s,,j,’
and if u’ = 1 (such galleries are ca.lled loops) we obtain the relation SjnL . si, j, = 1.
(2.4)
All relations in W follows from the basic relations which are of 2 types. The first one consists of relations s;.= 1
(2.5)
and the second type involves triples of indices 1
s23s13s12s23s13s12
-
(2.6)
It corresponds to the loop shows on Fig. la. Another way to write (2.6) is s23s13s12
=
s12s13s23.
(2.7)
The sides of Eq. (2.7) correspond, respectively, to the counterclockwise and the clockwise galleries leading from C, to C (see Fig. 1b). All these assertions hold (with slight modifications) for a wider class of groups called reflection groups (cf. [4] or [9]). We use the notation of Section 1 to define the Bethe Ansatz in the general case. So we assume that p 2 1, d> 1 and the matching operators A”(A, p), &,I, ,u) for the two-particle problem are given. Recall that the matching operators (sometimes we
BETHE
ANSATZ
AND
23’13
Xl <
27
EQUATIONS
YANG-BAXTER
si2s23s13s12c+
52
x3
gallery c, to c_ c+ X4<
X2
i.;;i ,‘23
rl
4 Cx3
X3
sl2
FIG.
l(a)
Loop
J
corresponding
to Eq. (2.6).
(b)
two galleries
corresponding
to Eq. (2.7 ).
28
EUGENEGUTKIN
call them matching E,=(@NCP)@Cd.
coefficients)
satisfy equations
(1.3))( 1.6). For N> 2 let
DEFINITION 1. Let N be any integer > 2 and let C(RN, EN) be the space of continuous functions on RN with values in E,. Let A”(& p), B(A, ,u) be a pair of matching operators. A functionfe C(RN, EN) is called a Bethe Ansatz state if the following two conditions are satisfied.
(1) For any w E W the restriction off on C,,. is a linear combination of plane waves. (2) Let C, and C,,. be two adjacent chambers and let h, be the wall between them so that C, corresponds to {xi d *xi> and C,,. corresponds to {xi 3 x,). Let flc, =c
(2.8
u(A) e”(.x),
where n = (2 ,,..., AN). Then flc,, =C [A(A;, Ai) u(A) e”(x) + &A,, A,) u(A) e~+‘(-~)l.
(2.9
Intuitively, a Bethe Ansatz statefis given by a linear combination of plane waves in every chamber and the expressions of f in adjacent chambers match. The problem is to characterize the matching coefficients which allow a full set of Bethe Ansatz states. DEFINITION 2. We say that the matching coefficients A(& p), &%, p) satisfy the full Bethe Ansatz if for any N> 2, almost any 2 = (A, ,..., 2,) E CN and any p E E, there exists a Bethe Ansatz state f such that
f 1c, = ue’(x).
(2.10)
3. We say that matching coefficients A(A, p), B(i, 11) satisfy the (antisymmetric) Bethe Ansatz if for almost any /i = (2, ,..., %N) E CN
DEFINITION
symmetric
p exp(i(fx+my))
A(l,m)p
exp(i(fx+my))+
B(l,m)p
exp(i(mx+ly))
I X
FIG.
2.
Matching
X’Y
coeflicients
for the Bethe
Ansatz.
BETHE ANSATZ AND YANG-BAXTER
EQUATIONS
29
(N 3 2) and any u E E, there exists a Bethe Ansatz state j’which is symmetric (resp. antisymmetric) in X, ,..., -yN and such that (2.1 I ) with ~(1, n)=u.
3. SCATTERING
OF THE TYPE ONE: FULL
BETHE ANSATZ
Recall that p = 1 and the matching operators A(;., p), B(i., p) act on C”. Assume that B(E.,p) (and therefore A(A, ,u)) is a meromorphic matrix function of (k ~0 and that A(& p) + 1 when 3.-p + co (these conditions are satisfied in applications, see Sect. 6). Then det A(L, p) is a meromorphic function which is not identically zero. PROPOSITION
1. Consistenq~conditions for the tnutching oprrtrtors ore ryuiwlen t
to (3.1 )
A ( j., p) + B( 2, 11)= 1 and B(p, 2) = -B(i,
(3.‘)
~1).
Proof: Since in our case A = 2, B = B, Eqs. ( 1.4) and (3.1 ) coincide. Equations (1.5) and (1.6) become B’(E., p) + A(p, 2) A()., p) = 1
(3.3)
A(;., ,u) B(L p) + B(p, j.) A(i. p) = 0.
(3.4)
and
Substituting A = 1 ~ B into (3.3) and (3.4) we obtain B’(I.,~)+(l-B(~,I))(I-B(I,~1))=1 and (1-B(LpO)
B(/1.~)+B(~1,i)(l-B(r.,~o)=O
implying
and -B’(k
P)- B(p, 1) B(L /A)= -B(&
p-
B(p, r,).
30
EUGENEGUTKIN
Hence (3.3) and (3.4) are equivalent. we have from (3.4)
Since B(A, II) commutes with A(1, p), by (3.1),
[BOYPL)+ B(P>A)1 A(& 11)= 0 and multiplying
(3.5)
(3.5) by A ~ ‘(A, p) ( recall that det A(& p) # 0) we obtain (3.2).
PROPOSITION 2. Let A(k p), B(l, 11)be a pair of matching coefficients. The full Bethe Ansatz holds if and only if the functional equations
Ah
v) A(A v) AtA PL)= A(A PL)A(& v) A&, VI,
A(k v) Ah, v) WA, cl) = B(A PL)A(k v) Ah
Ah
(3.6)
VI,
(3.7)
A(& v) A(& P) B(PL, v) = B(P, v) A(4 v) A(k r-l),
(3.8)
2) B(k v) A(& P) + B(A P) BbL, v) B(A PL) = A(v, P) @A, v) Ah, v) + B(P, v) B(A P) HP, v),
(3.9)
A(& PL)WL, v) 41, P) + B(P, 1) B(L v) 42, P) = B(k v) A(A P) B(P, v),
(3.10)
Ah
(3.11)
v) BbL, v) B(i, P) + B(v, P) WI-, v) Ah, v) = WA, v) Alp, v) B(A PL)
are satisfied, which are understood as identities for meromorphic operator functions A(., .) and B(., .). Proof:
Set N= 3 and let f be the Bethe Ansatz state such that flc+ = ,,Ji(1.‘;+u?.+\.I)
Take the counterclockwise gallery leading from C, to C~ = s,~ C, (see Fig. 1). Going along the gallery from a chamber to the adjacent one and using the matching equation (2.9) we determine the coefficients p(w, /1) of the expansion (A = (4 P? v)), (3.12) flc_ =Cu(w, A) e”“(x). II’ Doing the same for the clockwise gallery from C, to C another expansion f 1C_= C u(w, A ) e”‘“(x).
(see Fig. 1) we obtain (3.13)
The formulas for u(w, ,4) and v( w, /i) are obtained by straightforward computations parallel to those of [S, Sect. 4 (we spare the details). We have, for instance, u(l,A) = A@, v) AU, v) 42, pL)u
(3.14)
v(l, A) = A(& p) A(J4 v) A(/& v)u.
(3.15)
and
BETHE ANSATZ AND YANGBAXTER
EQUATIoNS
31
The full Bethe Ansatz holds for N= 3 if and only if for any MIE W and almost all A = (4 11,VI, u(w, A) = u(w, A ),
(3.16)
which gives 6 vector equations. Since u is arbitrary, (3.16) is equivalent to 6 operator equations (3.6)-(3.11). For instance, for w = 1 we have, by (3.14) and (3.15),
which is equivalent to (3.6). For any N> 3 relations in W, are implied by (2.5) and (2.7). Translating this assertion to the language of galleries and loops we see that any loop in RN is a composition of loops conjugate to that on Fig. la. Equations (3.6)-(3.1 I ) mean that the matching is consistent along any such loop, therefore along the composition of any number of such loops. Thus Eqs. (3.6)-(3.11) are equivalent to the consistency of matching along any loop for any N, i.e., to the full Bethe Ansatz. THEOREM 1. Let A(& p), B(1, p) be a pair addition to (3.1) and (3.2) the assumption that
of matching coefficients
satk[\?ng in
det B(j., p) # 0
(3.17)
WA,PL)= [F(n) - F(P)1 ‘,
(3.18)
identical1.v. Then
where F is a meromorphic operator-valued function such that F( E.) commute,fbr all i and det [F(n) - F(p)] is not identically zero. Conversely, if F is any such ,funr.tion then the matching coefficients given by (3.18) satisjj the full Bethe Ansat:. The following lemma, which is needed to prove Theorem 1, seems to be of independent interest. LEMMA 1. Let E be a complex finite-dimensional linear space, and let A(x, J,) he a meromorphic function on C x C with values in the space M(E) of operators on E. Assume that A(x, y) and A(z, u) commute if there are no more than 3 d&inct numbers in {x, y, z, u}. Then A(x, y) and A(=, u) commute for all X, J, :, and u.
Postponing the proof of Lemma 1 we now prove the theorem. Proof of Theorem 1. First we will show that Eqs. (3.6)-(3.8) are equivalent to the commutativity of A(;1, p) for all A, p. Equation (3.7) means that B(& 11) commutes with the product A(i, v) A(p, V)
32
EUGENE
GUTKIN
therefore, by (3.1), A(,I, p) commutes with A(& v) A(p, v) for any A, p, and v. From this and (3.6) we have
which we rewrite as
Cd& VI,44 VII A(& PL)= 0. Multiplying (3.19) by A ~~‘(II, p) we conclude that satisfies assumptions of Lemma 1. Therefore all satisfied. Conversely, if {A(i, p)} is a commutative obviously satisfied. Now the order of factors in (3.9))(3.11) does rewrite (3.10) as
(3.19)
the family {A( , )) of operators A(& p) commute and (3.8) is family then Eqs. (3.6k(3.8) are not matter and, using (3.2), we
44 PL)CHAPL)m v)-B(A PI HA v)-B(A VI m VII =a Multiplying
(3.20)
(3.20) by A(1,p) ~ ’ we get
CNA 11)+ m.4 v)l B(J.,v) = WA PL)m, v).
(3.21)
Dividing (3.11) by A(p, v) we also get (3.21), hence (3.10) and (3.11) are equivalent to each other and to (3.21). Rewrite (3.9) as
[A(& P) Ah A) - A(K v) A(v, ~11WA VI = B(A ~1 WL, v)CWp,v) - W, pIl(3.22) and use (3.1) to get rid of A in (3.22). After some simplifications
we obtain
[m4 vJ2- NJ”?PI’1 B(A v) = WA,PI m, VKm4 v) - B(4 PL)l and factoring Q,
v) - B(I, p) out we have
CQL, v) - B(k PL)IC(B(LP) + WP, v)) @A v) - WA ~1%A ~11=O, (3.23) which of course follows from (3.21). Thus the system of equations (3.6)-(3.11) is equivalent to the commutativity of matching coefficients and Eq. (3.21). Denote B(& CL))’ by G(& p) and multiplicity (3.21) by G(& p) G(p, v) G(& v). We obtain
(31, PL)+ G(P, v) = (32, v).
(3.24)
To work with this equation it is convenient to denote ;1, p, v, by x, y, z, respectively. Using (3.2) we rewrite (3.24) as G(x, z) = G(x, Y) - G(z, Y).
(3.25)
BETHE
ANSATZ
AND
YANG-BAXTER
33
EQUATIONS
Since &G(.K.
.I’): =
lim G(x+A, 1- 0
y+ A)-G(.u+A,
.v-G(-u, A2
y+A)+Gk
.v)
(3.25) implies that (3.76) The general solution
of (3.26) is G(x, y)=g(-~)+h(g)
and (3.25) implies that h(.~) = -g(s).
Denoting
g by F we have
G(.Y, y) = F(s) - F(y),
(3.17)
which is equivalent to (3.18). Of course, the function F(s) is defined by (3.27) only up to a constant operator. Since G(x, ~1) commute with each other, we can choose this constant so that the operators F(s) also commute. For any operator function F(x) satisfying the assumptions of Theorem 1 the function G(s. J,) given by (3.27) satisfies (3.24). therefore B( X, y) = G(.Y, J,) ’ satisfies (3.21) which implies (3.6)-( 3. I 1). This proves the theorem. Pm$c?f’ Lumm 1. Since the function A(x, J.) is meromorphic of points (s, J’), where A(s, 1’) is defined, is open and dense in the exposition we will argue as if R = C’“. A subspace F c E is an eigenspace of a set X of operators on there is a number i. (eigenvalue of .4 on F) such that (A - l)“F= any commutative family X of operators there is a unique direct E=xE, !El of E into eigenspaces of X such that if E,,(A) is the eigenvaiue ).,(A ) # i,(A) for at least one A E X. Fix .YE C and let E=x
E,(.Y)
on C’, the set R C’, and to simplify E if for any .A E ,I 0 for some H. For decomposition (3.28) of A on E, then
(3.29)
be this decomposition for the commutative family (A(.Y, ~2): J-E C)-. We denote by E.,(s, J,) the eigenvalue of ,4(x, ~1) on E,(x). For generic y we have j-,(.x, J) # j*,(.u, y). therefore for generic J’ the eigenspace E,(x) can only split if we move horizontally away from (.I-, y). Since dim E,(.K) 3 1 we conclude that there is a unique decomposition (3.28) of E
34
EUGENEGUTKIN
into eigenspaces of the set { A(x, y): x, y E C} and that for almost all (x, v) (call them regular) we have &(x, y) # Aj(x, y). Since now the assertion of the Lemma reduces to the same question in each E;, we can assume without loss of generality that 111= 1, i.e., A(x, y) = 1(x, y) + N(x, y), where N(x, y) are nilpotent and have the same commutativity properties as A(x, y). Subtracting J(x, y) off we reduce the question to the case when A(x, v) are nilpotent. By the same argument as above we show that E contains a nonzero subspace E, such that A(x, y) E, = 0 and E, = Ker ,4(x, y) for regular (x, y). We can now argue by induction on dim E and assume that the induced operators A r(x, y) on E, = E/E, commute with each other. Then the operators ,4(x, -v) obviously commute. The lemma is proved. Remark. Let us drop assumption (3.17) in Theorem 1. The argument of Lemma 1 shows that Cd= E,@ E,, where E, and E, are invariant under B(& p), the restriction B,(& p) = B(1, p)IE, satisfies (3.17) and B,(il, p)=B(& p)IE, is nilpotent for all 1 and CL.Equation (3.21) has plenty of nilpotent solutions. For instance, let dim E, = 2n and let &(,I, p) have the block structure
where C(A, p) is an arbitrary meromorphic function with valus in M(C”). (3.21) is satisfied. But if we require that B,(I1, cc)= 0 then Theorem 1 yields
Then
COROLLARY 1. Let A(& p), B(& ,a) be a pair of matching coefficients. Let Cd= E, @E, be the canonical decomposition guaranteed by Lemma 1 such that det[B(& p)lEOl =0 and det[B(l, p)lE,] #O. Assume that B(l>, p)JE0= 0. Then there exists a meromorphic function F on C with values in M(E,) such that F(n) all commute, det [F(A) - F(p)] # 0 and
COROLLARY 2. Let the matching coefficients depend on I-p, i.e., 44 PL)= A(A - ~1, B(A PL)= B(J. -PI and assumethat B,(13.- ,a) = 0. Then the full Bethe Ansatz is satisfied if and only if B has the form
B(1) = CA - ‘,
(3.30)
where C is an operator on Cd (E, = Ker C). ProoJ: Assume for simplicity
is trivial).
Then, by Theorem
of exposition
that E, = 0 (the extension to E, # 0
1, B-‘(l-p)=
F(%-
F(p).
BETHE ANSATZ AND YANG-BAXTEREQUATIONS
35
Setting p = 0 in the equation we have F(i)=
B-‘(I-)
hence F(I - p) = F(R) - F(p). Besides, by (3.21, B-‘( -2) = -B-‘(I.),
thus F(A) satisfies (3.31)
F(i.+p)=F(l.)fF(p). The general solution of this equation is F(i) = F. i,
where F is an arbitrary nondegenerate matrix. Setting C = F ’ we obtain (3.30). COROLLARY
3. Let a(],, p) and h(A, p) he a pair qf’scalar matching coeJficiPnts.
i.e., d= 1. (i )
The)! sati$v the fidl Bethe Ansatz tf and or&j if’
where g is anI> nonconstant meromorphic jknction. (ii) ff a and h depend on I. - ,u on!,+, then the ,full Bethe Ansut: is sati$kd if and only ij h(i,,u))=C(~-11)
‘.
(3.33)
bcherec is an arbitrary complex number. Proof: Assertion (i) follows from Corollary 1 since its assumption is trivially satisfied for d = 1. Assertion (ii) obviously follows from Corollary 2.
4. SCATTERING SYMMETRIC
OF TYPE ONE: AND ANTISYMMETRIC BETHE ANSATZ
Recall that p = 1, E, = C”, the action of W, on E,,,, is trivial and (2.1 ) becomes
THEOREM 2. (a) Any pair A(,l, ,D), B(,l,p) antisymmetric Bethe Ansatz.
oJ1 matching coefficients satisfks
(b) Let A(R, p), B(,l, p) he a pair qf commuting matching co@cients. Then the symmetric Bethe Ansat; is satisfied.
36
EUGENE
GUTKIN
Proof It is convenient to change notation Bethe Ansatz state fin a chamber C,. as
slightly and write the expansion of a
flc, =I 4~ A) e”(x),
(4.1)
A
where the summation We have
is over all /i belonging to a W-orbit.
Denote u( 1, /i) by u(A).
flC,(4=flC+w’x)
(4.2)
flc,(X)=(-)‘~lc+(~~~‘~~)
(4.3)
and
in the symmetric and antisymmetric case, respectively (where (- )“‘ means det w). From (4.1), (4.2), and (4.3) we immediately obtain u(w, A) = u(u’-‘A)
(4.4)
u(w, A) = (- )“‘u(w’A)
(4.5)
and
in the symmetric and antisymmetric cases, respectively. Let C,. and C, be two adjacent chambers and let s = sij be the reflection about the wall between them where C,,. corresponds to x, < xi. Consider the coefficients u(w, A), u(u’, sn), u(u, A), and u(u, sn), where n = (A,,..., AN). By (2.9), (4.4), and (4.5) we have u(u, A)=
fu(w, s/l),
u(u, s/l) = +qw,
(4.6)
A),
(4.7)
u(u, A) = A(l*;, /Ii) U(W, A) + B(A,, ni) u(w, s/i),
(4.8)
u(u, s/l) = A(A,, Ai) U(W, s/f) + B(&, II,) u(w, A),
(4.9)
where the plus sign (resp. minus sign) corresponds to the symmetric (resp. antisymmetric) case. Solving (4.6) for u(w, sn) and substituting into (4.8) we obtain after elementary transformations [l
fB(Aj,
where now minus corresponds obtain from (4.10)
ni)]
U(U,
A)=A(A,,
A,)
to the symmetric
A(A,, ;1,) u(“, A)=A(ni,
Aj)
U(Wy
A),
(4.10)
case. Using (3.1) and (3.2) we u(w, n)
(4.11)
BETHE
ANSATZ
AND
YANCBAXTER
37
EQUATIONS
and (4.12)
A(&, A,) u(u, A) = A(E.,, R,) zf(M’, A)
in the symmetric and antisymmetric (4.11) and (4.12) are equivalent to
cases, respectively. Since A(& ,u) is invertible,
II(L), A) = A ‘(A,, A,) A(/?,, I.,) tf(w, A)
(4.13)
u(v, A) = U(M., A).
(4.14)
and
respectively. NOW solve (4.6) and (4.7) for U(U, /1) and U(L),sA), respectively, and substitute into (4.8) and (4.9). We obtain [k 1 - B(i,, ii)] u(M~,sA)=A().,,
(4.15)
i,)u(ti’, A)
and [ fl
- B(A,, A,)] u(M’, A)=A(l.,,
which are equivalent since /i is arbitrary From (4.15) we get
i,) u(tt‘, .x/l),
(plus corresponds to the symmetric case),
u(M’,sA)=A-‘(~~,~.~)A(~“,,~.,)u(I~.,
(4.16)
A)
and (4.17)
u( h(‘, s/i) = -u( M’, A )
in the symmetric and antisymmetric cases, respectively. The Bethe Ansatz statefcorresponding to u E C” and A, E CN exists if and only if the system (4.6)(4.9) of homogeneous linear equations (where M’, u E W and II = g/i,. gg W) has a solution u(g, A) with U( 1, A,) = U. Equations (4.6).-(4.9) are equivalent to (4.13) and (4.16) in the symmetric case and to (4.14) and (4.17) in the antisymmetric case. The obvious solution of the latter system is u( g, wAo) = ( - )“‘u,
(4.18)
which settles the antisymmetric Bethe Ansatz. If A,, is regular, i.e., IVY,= A, implies HZ= I, then Eqs. (4.13) and (4.16) are equivalent. To show the existence of Bethe Ansatz states it is convenient to work with the system (4.13). This system has a solution if and only if for any relation (2.4) in W we have (/iO = (J ,,..., A,)) A ‘(i.,,l.in)A(i,n,
n,~)...A-‘(~,,,i.,,)A(l.,,,
i,,)=
I.
(4.19)
38
EUGENEGUTKIN
It suffices to check (4.19) for the basic relations (2.5) and (2.6). Equations (4.19) corresponding to relations (2.5) are trivially satisfied. Replacing (2.6) by (2.7) we rewrite the corresponding equation as
which is obviously theorem is proved.
satisfied if the matching
coefficients A(5 ,u) commute.
The
COROLLARY 4 (of the proof). Let A(& p), B(I, p) be any pair of matching coefficients. They satisfy the symmetric Bethe Ansatz if and only if Eq. (4.20) holds.
Remark. The argument of Theorem 2 also answers the following question. We say that the weak form of Bethe Ansatz holds if for almost all A = (A, ,..., AN) there is a u # 0 which determines the symmetric Bethe Ansatz state (2.11). Denote by A(IZ,, 1,, &) the left-hand side of (4.19) corresponding to the relation (2.6). COROLLARY 5. Matching coefficients A(& ,u), B(1, ,u) satisfy the weak form of symmetric Bethe Ansatz if and only if the equation
det[A(L, CL,v) - l] = 0
(4.21)
holds. ProoJ: By the argument of Theorem 2, the weak form of symmetric Bethe Ansatz holds if and only if for almost all (A, ~1,v) there exists u # 0 such that A(n, ,u, v)u = u which is equivalent to (4.21). We will calculate the expansion (2.11) of symmetric Bethe Ansatz states. First introduce some notation. A gallery C, = Co, C, ,.,., C, = C, leading from C, to C, defines the sequence (iI, jI) ,..., (i,, j,) of orde re d pairs of indices given by the rule that C,_ i is contained in the halfspace (xi, < xj,}. The ordered pairs (i, j) correspond to the roots y of the symmetric group W (cf. [4]). We denote the situation described above by u-v Y,s...,Yn
(4.22)
and call y1 ,..., Y,, the gallery sequence of roots. If y = (i, j), denote the reflection sii by s,, and set -y = (j, i). If y = (i, j) and A = (2 ,,..., A,,,), denote &- lj by (y, A) and set A(Y, A) =A(ni, Aj). The roots y = (i, j) with i< j are called positive (y > 0) and the roots ai= (i, i+ l), i= l,..., N - 1, are called simple. Reflections 0; = s~,~+i, i = l,..., N- 1, are called simple. A decomposition W=o,“‘cT”
(4.23)
BETHE
ANSATZ
AND
YANG-BAXTER
39
EQUATIONS
of WE W into a product of simple reflections is called reduced it it has minimal length. Denote by y + wy the natural action of W on the roots. Let (4.23) be any decomposition and let a,,..., c(, be the corresponding simple roots. Consider the sequence u,, CJ~~~-, ,..., (T,. . . c, = IV-’ of group elements and the sequence y, = a,,, y2 = ~,,a, ~, ,..., yn = (o,, . az) 01~of roots. Then 1-w
I
(4.24)
y, . Yn
and (4.23) is reduced if and only if y ,,..., yn>O. The roots y=y ,,..., y,, in (4.24) depend only on w and they are characterized by y > 0, MT?< 0 (i.e., - M’Y> 0). The assertions above hold for general reflection groups and the reader can find proofs in [4] or in [7,9]. THEOREM 3. Let A(A, p), B(l., p) be commuting matching operators and ,fin u E Cd, A E CN denote by f,,,* the symmetric Bethe Ansat: state which has the e.upansion
fu.,,IcL =Cu(wA)e”“(x)
(4.25)
ulith u(A) = u (by Theorem 2, f,,,, exists ,for all II and almost all A ). Then u(wA)= [I-I.ico A(;‘, A)]-‘[n,,, A(y, wA)lu, i.e., (4.26) ProoJ: The pairing
(y, A) is invariant
with respect to the action of W:
( y, A ) = (WY, tt,/l ).
(4.27)
Let (4.23) be a reduced decomposition and let (4.24) be the corresponding gallery sequence. Set u,, = g,,, U, ~, = o,,cn , ,..., U, = CJ,, (T, = w ‘. The chambers UsC’ +. C + are adjacent for k = O,..., n - 1. and uk+l Let z be a simple root, let s be the corresponding reflection and let VE W. From (4.16) setting u = 1 and replacing n by VA we have zc(svA)=A~~‘(-r,vA)A(a,vA)u(vA)=A
Applying steps
‘(-0
(4.28) successively to 0,~’ ‘..o,,A, u(wA)=A
‘(-u,M,,
A) A(uzul, A)...
‘cr,A)A(v
oza,...~,z~
‘x,A)u(vA).
(4.28 )
,..., o,,A we obtain after II
A ‘(--a,,. A) A(cz,,, A) u(A).
(4.29)
The roots ?,I = a,,..., y, = uzcyl form a gallery sequence (4.24) and, since (4.23) is
40
EUGENEGUTKIN
reduced, they are characterized by y>O, wy ~0. Using that A( -y, A)= A(y, -A), that A(, ) commute and that u(A) = u we rewrite (4.29) as u(wA)=
n
A-‘(y,
-A) A(y, A)
u.
(4.30)
I
YSO
WY<0 Set A(w,A)=
(4.31)
n A(y,A). y>o ny
Then (4.30) can be written as fu,/,Ic.+ =I
II’
A(w, A) A-‘(w,
-A)
u&““(x).
Let wo: 1 --) N, 2 + N - l,..., N -+ 1 be the longest element multiply (4.32) by A(wo,
-A)=
(4.32) of W and let us
n A(y, -A). YSO
(4.33)
We have for any M’E W, A(w, A) A-‘(w,
-A) A(w,, -A) (4.34)
= n A(Y, A) n NY, -A). i’>o v>o wg -c0 ““i > 0 Using that A(y, -A) as
= A( -y, A) and (4.27) we rewrite the right-hand
side of (4.34)
fl A(Y, -WA), y>o which means that
u(wA)= n Ah,-wA)lA(y, -A)= n A(Y,wA)lA(y,A) y>o
(4.35)
Y
and proves the theorem. COROLLARY 6 (of the proof). Let A(& p), B(2, 11) be commuting matching coef ficients. The symmetric Bethe Ansatz state fu.n exists for A = (A,,..., I,) if the operators A(n,, Azi)are definedfor all i # j and det A(Ibi, A,) # 0 for i < j. It is giuen by
fu,~lc+Ew~w[
n A(y,/f)lA(y, -~)]~~Y.~).
Y>O M.y
(4.36)
BETHE
ANSATZ
AND
YANG-BAXTER
41
EQIJATIONS
Proof: Formula (4.36) follows immediately from (4.30) obtained in the proof of Theorem 3. It makes sense if the operators A(y, /i) are defined for all 7, that is A(L,, I.,) exist for i #j, and the operators A(y, -,4) are invertible for 1’ > 0, that is, det A(L,, L,)#O for i~l’ ‘(j)l is the length ofn’. THEOREM 4. (a) Let A(i, p), B(J., ,LL)be cornrnuting matching coqfficients tl,ith values in M(C’). For A = (1, ,..., 2,) and u E C” the s~wmetric Bethe Ansat: stute f;,, , is given by the equivalent e.upressions
n A(i,,,. I(,,, E.,,.-+,,)/A(jU,. i.,)
I(,
A().,, ).,)/A(;.,.
II
;,)
ue” ‘(.Y)
(4.37)
ue”.‘(.‘i),
(4.3X I
1
‘(i)
'(il>bt
1
Lchere the summation is over all perrmtations 11’of 1l..... N) (b) Let the nlatching coefficients A()., p). B( A, p) he given bus B(j,, p) = C(3. -p)
(4.39)
‘.
where CE M(C”). Then the s~.wmetric Bethe Amutz stute f;,. , e.xisf.s fiw uimo.vt till A = (2, ,..., &,,) and is given by the equivalent e.upre.s.sion.s
und .Ll
c+ = C det 11’
,I’
n
I
c-(&-a,, C-(i,-2,)
ue”‘,‘(s).
(4.41 )
Proof Formulas (4.37) and (4.38) are straightforward translations into the language of permutations and ordered pairs (i, .j) of (4.26) and (4.36), respectively. Formulas (4.40) and (4.41) are derived from (4.37) and (4.38), respectively, by substituting A(k
p) = [(1.-p)
- C]/(l. - p).
42
EUGENEGUTKIN
COROLLARY 7. Let A(& u), B(I, p) satisfy the full Bethe Ansatz and depend on I- p. Then the symmetric Bethe Ansatz state f,,n exists if lj - Izi is not an eigenvalue of C for all i< j and it is given by (4.40) or by (4.41). Proof By Corollary 2, B(& p) has the form (4.39) therefore fu,n exists for almost all /i and is given by (4.40) or, equivalently, by (4.41) if the formulas make sense, which they certainly do if det( C- (lj - A,)) # 0 for i < j. 5. FULL BETHE ANSATZ AND GENERALIZED YANG-BAXTER
EQUATIONS
In this section we set p > 1, d = 1, and study the case (ii) of Section 1, i.e., scattering between particles of different kinds. We recall from Section 1 that the matching operators are T(2, p) and R(I, p) and they act on Cp@ Cp = Cp2. They satisfy the consistency conditions T(4 CL)+ R(A P) = 1,
(5.1)
R*(k P) + Tbu, A) T(A P) = 1,
(5.2)
and T(L P) N4
P) + NCL, 1) W, 11)= 0.
(5.3)
As in Section 3 we assume that the matching coefficients depend meromorphically on Iz and p. We no longer assume that A(& p) + 1 when i -CL + co since this is not so in many applications. The cases T(& p) = 0 or R(A, p) = 0 identically are trivial and we exclude them in what follows. Proposition 1 in the present context becomes PROPOSITION 3.
Assume that det T(2, p) # 0 identically.
Then
N/4 A) = -Nil, P).
(5.4)
The space E, of Section 2 is equal in this case to ON Cp and W, acts on ON Cp by permutations of factors. Now Definitions 1, 2, and 3 of Section 2 apply in the present case. Let A be an operator on Cp 0 Cp. For 1 < i < j d 3 we denote by A, the operator on Cp@ Cp 0 Cp which is given by A on the product of the factors i and j in the tensor product and is equal to identity on the remaining factor. For any 1 < i < j< 3 the correspondence A + A, is an isomorphism of M(CP@ Cp) into M(CP@ CP@ cq. THEOREM 5. Let T(2, p) and R(A, p) be a pair of matching operators for the twoparticle scattering (p > 1). Then the full Bethe Ansatz holds if and only tf the matching operators satisfy the equations T&t T23(4
v) T,,(A v) v) T&u,
v)
T,2(4 R,2(4
v) =
T,,(k
PU)T,,(k
v)
T23(4
v),
(5.5)
P) =
R12(4
PL) T,,(L
v)
T23bL,
~1,
(5.6)
43
BETHE ANSATZ AND YANGBAXTEREQUATIONS
= R,,(v, P) R,,(A v) TAP, v) + T&L, v) R,,(k P) MP,
(5.8)
~1,
R,,(P, 1-1Rn(A v) T,,(A P) + T,,(A cl) R,,(P> v) R,z(L PL) = R,,(k v) T,,(k PL)R&L, v), T,,(P~ 4 R,AL v) T,AA PL)+ RAA
(5.9) PL)R,,b,
= T,,(v, cl) R,,(A v) TAP, v) + R&L
v) Rdj., P)
v) R,,(k PI R&,
VI.
(5.10)
Proof Equations (5.5)-(5.10) are obtained by matching the Bethe Ansatz states for three scattering particles. They are the analogs of Eqs. (3.6)-(3.11) in the present situation and the proof of Theorem 5 is completely analogous to the proof of Proposition 2. We spare the details. COROLLARY 8. Let the matching coefficients To., p) and R(,?., p) depend on the difference 1. - p. Then the full Bethe Ansatz is satisfied if and only if the equation.7
T,,(u) R,du + uV’,,(u) = T,,(u) T,,(u + u) T,,(u),
(5.11 )
Tzdu + u) T,&u) R,Au) = R,Au) T,,(u + 1,) TJu),
(5.12)
R&u) T,,(u + u) T,,(u) = T,,(u + t’) T,,(u) RX(U).
(5.13)
R,,(u + u) T,,(u) R,,(u) = Rd-0)
R,,(u) T,,(v) + T,,(u) R,,(u) Rzdo),
R,,( -u) R,,(u + u) T,,(u) + TAu)
(5.14)
R,,(u) R,z(u) (5.15)
= R,,(u + 0) Tutu) R,,(u), Tzd -u) R,,(u + u) T,,(u) + Rn(u) R,,(u) R,z(u) = T,,( -u) R,,(u+ 0) T,,(u) + R,z(t;) R,,(u) R,,(u)
(5.16)
hold. Proof. Equations (5.1 lt(5.16) are obtained from (5.5)-(5.10) by substitution A-p=u, p-v=u, E,-v=u+u. Equation (5.11) for the operator function T(u) is known in the literature as the Yang-Baxter equation. Recently it came up independently in the work on factorizable S-matrices [ 193 and on the soluble models of statistical mechanics [ 11. In view of this we propose to call the system of equations (5.5)-( 5.10) or (5.11)-( 5.16 ) the generalized Yang-Baxter equations. The following is immediate. COROLLARY
9.
Let
the matching
operators
T(A, p), R(A, ,u) depend
on A-p.
[f’
44
EUGENEGUTKIN
the many-particle scattering problem has the fill Yang-Baxter equation (5.11).
Bethe Ansatz then T(u) satisfies the
6. EXAMPLES Case (i). We will consider singular differential operators on P-valued on RN of the form G = -d + singular terms
functions (6.1)
where A= 2
a2/ax;
(6.2)
i=l
is the standard Laplacian and the singular terms are delta-function-type singularities supported on the hyperplanes {xi = xi). An operator (6.1) is the positive Laplacian with boundary conditions on {xi = xi, i, j = l,..., N} determined by the singular terms in (6.1) (see examples below). These boundary conditions define a pair A(& p), B(L, p) of matching coefficients satisfying the conditions (3.1) and (3.2). The corresponding Bethe Ansatz states (see Definition 1) are called the Bethe Ansatz eigenstates of G. They are in fact the generalized eigenfunctions of G since they satisfy -Af=Ef (6.3) and the boundary conditions imposed by G. We say that the operator (6.1) satisfies the full (resp. symmetric) Bethe Ansatz if the corresponding matching coefficients satisfy it (see Definitions 2 and 3). EXAMPLE
1 (Delta-potential).
Let C be any d x d matrix and set
H= -A+2C
1 &xi-xi).
(6.4)
i-zj
Denote by a/ax, - a/ax,)fl+ the jump of the “normal derivative” (a/axj - a/ax,)f on the hyperplane (xi = xi}. It is standard to check that the boundary conditions imposed by (6.4) are - a/ax,)fI f = 2cf
(6.5)
for l
to (6.5) consider the case of
(a/axi
(6.6)
BETHEANSATZ
AND YANG-BAXTER
45
EQUATIONS
The left-hand side of (6.5) is the restriction where i= fi. i(~ _ ~) Uel(;..,+ll.,‘)+i(j”-~)A(~,~)uei(‘-~+~““+i(~-i)B(~,IOue”~”+”’
to r = .Y of which
gives, taking (3.1) into account. 2i(p - 2) B(A, p) ue”L+l”r.
(6.7)
The right-hand side of (6.5) is 2C’ue”” + I”’ hence B(/l,~)=C/i(~c-it)=iC(j.--~)
‘.
(6.8 1
Thus the operator (6.4) satisfies the full Bethe Ansatz for any matrix C. Hence the symmetric Bethe Ansatz is satisfied and denoting by ,/;,, , the symmetric Bethe Ansatz eigenstate of H corresponding to II E C” and iI = (E,,..... E,L ) we have from (4.40)
with the summation over all permutations M’of i l...., IV;. Operator (6.4) is selfadjoint if C* = C and in the scalar case ti= 1, C is a real number, H is the famous Hamiltonian with delta potential (cf. [2, 14, 13, 18, 61). EXAMPLE
2 (Delta shift).
Let C be any clx ti matrix and set
L = A + 2c 1 6(x; - .u,)(?/?s, + C/?-Y,1. I< ,
(6.10)
The boundary conditions imposed by (6.10) are (d/Ss, - (‘/&Y,,fl * = -2C( ?/?s, + ?/?.u,),f:
(6.11 )
Computing the matching coefficients as in Example I we get the equation
which implies
By Corollary 2 to Theorem 1, the full Bethe Ansatz is not satisfied unless C = 0. However, by Theorem 2(b), L satisfies the symmetric Bethe Ansatz for all C. From (6.12) we have, by (3.1),
46
EUGENEGUTKIN
Substituting (6.13) into (4.37) we obtain a formula for the symmetric Bethe Ansatz eigenstate fu,n of L. The product in (4.37) becomes (we use summation over w instead of W-‘)
n [(n,(j)-n,(j))-(n,(j)+n,(i))C](S-ni)
icj
C(~j-ni)-(~j+ni)c](n,,j,-n,,i,)
=(-)“’
(‘w(j)
n
-‘W(i))
-
(l,(j)
+
n,(i))C
(6.14)
(A,j-Ai)-(lLj+Izi)C
i-cj
’
Using that
II (‘W*(i)+ ‘w,(j))= n (Ai + Aj), icj
i
we see that (6.14) is equal to
(-y& +n.,(j)-C]y[g-c]~ A,(j)
-
in(i)
I
41)
J
therefore
x explfi
(L(,,xl + ... +&v~X~)I.
(6.15)
In the scalar case d= 1 and C is a real number, L is the generating operator of the semigroup describing the motion of N Brownian particles with delta-shift. This operator is connected with the Burgers’ equation [ 151 and the Bethe Ansatz for L was used in [ 10, 1 l] to linearize the equation. EXAMPLE
arbitrary
d
3 (General delta-interactions). Fix x d matrices. Consider the operator
G= -d+2
n 30
and
let
Co,..., C,
be
C d(Xi-Xj) i
X [Co-C,(d/dX;+d/dXj)+
The boundary conditions (a/axi -
a/aXj)fl
’ ’ ’
+
( -
)“C,(a/CTXj
+
a/aXj)“].
(6.16)
imposed by G are +
=2[C,-C,(a/axi+a/axj)+
. . . + ( - )“C,( d/3x, + d/dx,)“]f:
(6.17)
BETHE ANSATZ AND YANG-BAXTER
Computing
the matching
coefficients as before, we get the equation
2i(p-i)B(I,p)=Z[C,--i(R+p)C,+
47
EQUATlONS
(i = \/ - 1 ) (6.18)
... +(-i)“(i+p)“C,,],
from which we have B(j.,~f)=[iC,+(;!+~)C,+
... +(-i)“~‘(3.+~)“C,,](E.-~O
‘.
(6.19)
In view of Theorem 1, G satisfies the full Bethe Ansatz if and only if C, = ... = C,, = 0, thus delta-potential is the only delta-interaction satisfying the full Bethe Ansatz. If the operators Co,..., C, commute, formula (6.19) shows that all B(E,, ,LI) commute, therefore, by Theorem 2(b), G satisfies the symmetric Bethe Ansatz. Assume that C,,,..., C,, commute and set y(.u)=iC,+sC,
+ “. +(-i)”
‘X’T,,.
(6.20)
Then from (6.19). by (3.1), we have A(i,p)= and, by computations
[(3.-~)-4(E.+~)]:(E.-~f),
(6.11 )
parallel to those of Example 2, we have
x exp[JZ
(&,.,,,.u, +
+ &,.,.,.y,,,)].
(6.22)
Case (ii). Examples of solutions of the Yang-Baxter equation are well known (cf. [12, 191) although the general solutions seems to be out of reach. We will return to the study of solutions of the generalized Yang-Baxter equations (5.5)-(5.10) and (S.ll )-(5.16) in a future publication.
REFERENCES I. R. J. BAXTER, “Exactly Solved Models in Statistical Mechanics. Academic Press. New York, 19X2. 2. F. A. BEKEZIN. G. P. POHIL, AND V. M. FINKELBERG,Schroedinger equation for the system of oncdimensional particles with point interaction. Vestnik Moskor. Unio. 1 (1964), 21. 3. H. BETHE. Zur Theorie der Metalle, 1. Eigenwerte und Eigenfunktionen der Atomkette, Z. P1zy.v. 71 (1931). 205. 4. N. BOURBAKI, “Groups et algtbres de Lie.” Chap. 4-6. Hermann, Paris. 1968. 5. M. GAUDIN. “La fonction d’onde de Bethe,” Masson, Paris, 1983. 6. E. GUTKIN, Integrable systems with delta-potential, Duke Murh. J. 49 (1982). 1. 7. E. GUTKIN, Operator calculi associated with reflection groups, Duke Marh. J. 54 (1987). I-18. 8. E. GUTKIN. Integrable many-body problems and functional equations, J. Marh. And. Appl.. in press. 9. E. GUTKIN, Geometry and combinatorics of groups generated by reflections, Enwigtz. Muh. 32 (1986). 95.
5Y5’176. I -‘I
48
EUGENE GUTKIN
10. E. GUTKIN AND M. KAC, Propagation of chaos and the Burgers equation, SIAM J. Appl. Math. 43 (1983), 971. 11. E. GUTKIN, Propogation of chaos and the Hopf-Cole transformation, Aduun. Appl. Math. 6 (1985), 413. 12. P. KULISH AND E. K. SKLYANIN. Solutions of the Yang-Baxter equation, J. Soviet Marh. 19 (1982). 13. E. LIEB AND W. LIENIGER, Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Whys.Rev. 130 (1963), 1605. 14. J. B. MCGUIRE, Study of exactly soluble one-dimensional N-body problems, J. Math. Phys. 5 (1964), 622. 15. H. P. MCKEAN, JR., Propagation of chaos for a class of nonlinear parabolic equations, in “Math. Studies,” Vol. 19, pp. 1777194, Van Nostrand, Princeton, NJ. 16. B. SUTHERLAND, An introduction to the Bethe Ansatz, in “Proceedings of the Winter School on Exactly Solvable Problems, Panchgani, India, 1985,” pp. l-96, Lecture Notes in Physics, Vol. 242, Springer-Verlag, Berlin. 17. L. A. TAKHTAJAN, Introduction to algebraic Bethe Ansatz, in “Lecture Notes in Physics,” Vol. 242, 175-220, Springer-Verlag, Berlin. 18. C. N. YANG, Some exact results for the many-body problem in one dimension with repulsive deltafunction interaction, Phys. Rev. Letr. 19 (1967), 1312. 19. A. B. ZAMOLODCHIKOV AND AL. B. ZAMOLODCHIKOV, Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models, Ann. Phys. 120 (1979) 253.
OF PHYSICS
176, 2248
(1987)
Bethe Ansatz and the Generalized Yang-Baxter Equations EUGENE GUTKIN* Department
of Marhemarics, Los Angeles,
Received
September
University California
20, 1985; revised
of Southern 90089 October
California.
17, 1986
INTRODUCTION
Bethe Ansatz plays an important role in the mathematical physics (see [S] and review [16]). Starting with the work of Bethe [3], Bethe Ansatz became an indispensable tool for solving various models of mathematical physics. It suffices to mention the classical papers [Z, 13, 14, IS] on Bethe Ansatz for the delta Bose gas, Baxter’s papers on Bethe Ansatz in statistical mechanics (see [ I] and references therein), the work of Leningrad’s school on Bethe Ansatz in quantum field theory (see, e.g., [ 173 and references therein). Obviously, the wide applicability of Bethe Ansatz is due to the fact that Bethe Ansatz in different fields means different things which are connected with each other in many ways. In this article we investigate the Bethe Ansats in the nondiffractive (nonrelativistic) quantum many-particle scattering. This framework goes back to the paper [2] on the scattering of particles with delta interaction. Our point of view is as follows. Suppose we have a system of N particles in one dimension which interact only at the moments of collisions. Thus the wave function of the system on the space R” = {x, ,..., xN} should be the same as the wave function of the free (noninteracting) system outside the union of “hyperplanes of collision” {-xi = x,, 1 < i < j 6 N}. These hyperplanes divide RN into N! disconnected chambers C,,. labeled by the elements w of the permutation group W,. In each C,. the wave function f of an eigenstate of the system should coincide with in eigenstate of the free system, i.e., a linear combination of plane waves. These special many-particle eigenstates are the Bethe Ansatz wave functions f: Transition rules from a chamber to another chamber should be the same for all Bethe Ansatz eigenstates J They are given by the matching operators (coefficients ,4(n, p), B(;1, p) (see Fig. 2) which are determined by the two-particle interaction (Hamiltonian) of the problem. Thus the physics of the problem determines these matching operators X4(,I, p), B(1, p), which is, essentially, the two-body scattering. Now the mathematical * Partially
supported
by NSF
Grant
DMS
86-00350.
22 0003-4916/87
$7.50
Copyright CI 1987 by Academic Press, Inc. All rtghts of reproduction in any form reserved.
BETHE ANSATZ AND YANG-BAXTER
EQUATIONS
23
properties of the matching operators shouid determine whether the Bethe Ansatz holds for the N-particle eigenstate for all N> 2. This determination is the subject of the present work. Our framework is sufficiently general which allows us to consider the scattering of particles of many kinds and of many external energy levels. We denote the number of kinds of particles by p and the number of external energy levels by d. We also consider the scattering of particles of arbitrary statistics (the full Bethe Ansatz). Our starting point is the observation that the full Bethe Ansatz holds if and only if the coeflicients A(L p), @I., II) satisfy a special system of functional equations. The general case reduces to the two special cases: (i) (ii)
p= 1, r/a I and p> 1. d= 1.
The case (i) is much simpler. We find the general solution of the system of functional equations (Theorem I), i.e., the general expression for .4(/., p), B(E..,I(). which satisfy the full Bethe Ansatz. Then we investigate the Bethe Ansatz for the Fermi and Bose statistics. It turns out that for the Fermi statistics the Bethe Ansatz is trivially satisfied. The Bose case is more interesting. We show that the Bethe Ansatz is satisfied if the matching operators commute with each other for all values iv, p of the wave numbers and find explicit formulas for the Bethe Ansatz eigenstates (Theorems 2-4). Thus for the many-particle scattering of type (i) we solve the Bethe Ansatz problem completely. In Section 5 we write the system of functional equations on the matching coefficients which is equivalent to the Bethe Ansatz in case (ii), i.e., the scattering of many kinds of particles. This system of six equations contains, in particular, the famous Yang-Baxter equation. The Yang-Baxter equation comes up in the study of the factorizable S-matrices [ 191 and the soluble lattice models of statistical mechanics [I]. Many solutions of the Yang-Baxter equation are found in the literature (cf. [ 121). In view of the above we call our system of six functional equations the generalized Yang-Baxter equations. We postpone a detailed study of these equations to another publication. In Section 6 we illustrate the results by a few examples.
1. TWO-PARTICLE
SCATTERING
A particle is characterized by its position s, its kind i and its external energy level k. Assuming that we have p kinds of particles and d external energy levels we consider the linear space CP generated by the basic vectors r,, 1 B id p, and the space C” generated by sk, 1 ,< k
expressions f(.)c)e Q s, i.e., the space of such states is the tensor L,(R)QCPQC?
product
24
EUGENE
GUTKIN
Since the general energy levels are the same for all particles, the two-particle states f(x, y) ei@ ej@ sk live in the space L,(R2) @ (Cp @ CP) 0 C4 The particles with positions at x and y do not interact outside the collision line {x = y}. Thus we look for the eigenstates of the two-particle Hamiltonian in the Bethe Ansatz form: exp(fi (Lx + py)u 0 u, where I and p are the wave numbers, u E CP 0 Cp, u E Cd. The interaction line {x = y} divides the xy-space into the two noninteracting regions {x > y } and {x < y }. A Bethe Ansatz eigenstate f with wave numbers L and p given by .f=exp(fi(Ix+py))uOr
(1.1)
in the region {x < y}, transforms into f = exp(fi
(2-x + w)) &A P)(U 0 u)
+ exp(fi
(CLX+ 2~)) &A P)(U 0 u)
(1.2)
in the other region {x > y}. The operator-valued functions A”(2, p) and &,I, p) are called the matching operators for the two-particle scattering. Note that A”(L, p) and &A, p) operate on the space CP 0 Cp 0 Cd. They are determined by the two-particle Hamiltonian or, equivalently, by the boundary conditions for the eigenstates. As a result of collision of two particles i and j which were in the state k we obtain particles i’ and j’ in the state k’. The transition k + k’ does not depend on the kinds of particles, therefore the matching operators must be of the form A”(4 11)= T(A PL)0 AtA 111, m
P) = RCA 11)0 B(k
PI,
(1.3)
where T, R operate on Cp 0 Cp and A, B operate on Cd. Continuity of the eigenstates f across the collision line implies A”(il, p) + B(A, p) = 1.
(1.4)
If the matching operators are consistent, matching backwords, from (x > y} into {x < y } we must recover the original eigenstate (1.1). The consistency conditions are B2(A, p) + A”@, 1) ‘qn, p) = 1
(1.5)
A(& p) B(ll, p) + B(p, 1) &A, ,u) = 0.
(1.6)
and
By (1.4) A(,?, p) and B(1, p) commute and, in view of (1.3), T(A, p) commutes with R(A, p) and A(1, p) commutes with B(1, p). Assume that d> 1 and let uk,
BETHE ANSATZ AND YANG-BAXTER
EQUATIONS
35 -_
1 6 k d d, be the common eigenvectors of A and B with the eigenvalues Us and h, Applying (1.4) to the vector u@ ilk and using (1.3), we obtain (a,T+h,R)l(O~,=uOo,,
(1.7)
which implies a,TSh,R=
1
(1.X)
for all h-. Assume that not all eigenvalues ah are equal or that not all h, are equal. for concreteness, (a,, h, ) # (a?, h,). Then (a,-a,)T+(h,
-h,)R=O,
(l.(J)
which, together with (1.8), implies that T and R are scalar operators. Denoting these scalars by t( A, p) and r( I., p) we have from ( 1.3) (1.10) That is, up to a normalization, T= R = 1, there is no interaction between different kinds of particles and the particles of each kind scatter on each other by the same rules. Thus we can assumewithout loss of generality that there is only one kind of particle. Analogously, assuming that p > 1 and that the matching operators T, R are not scalar, we come to the conclusion that there is no scattering between various energy levels. thus we can assumewithout loss of generality that d= 1. To summarize, the general many-particle scattering problem set up above reduces to the following two cases. (i) Particles of one kind, p = 1. the scattering is between various external energy levels. (ii) Particles of many kinds in the absence of the external field, d= 1. The scattering is between the different kinds of particles. In this article we study in detail the case (i) which is the scattering of particles of one kind. For brevity, we call it the scattering of type one. Case (ii) which is the scattering between different kinds of particles is considered in Section 5.
2. BETHE ANSATZ
Let N 3 2 and let W, be the permutation group of [ l,..., Nj. Let E, be a finitedimensional vector space over C with an action P + 11~of IV,. The group U’, naturally acts on RN by x -+ WY and on the space C(R,‘, EY) of continuous functions on RN with values in E, by (yf)(s)
= wf’(w ‘.Y).
(2.1)
26
EUGENE
GUTKIN
We denote elements of RN by x=(x ,,..., x,,,), y = (yl ,..., yN) and let (x, y) = CyI, x,y, be the scalar product. For 1= (2, ,..., ~,,,)EP denote by e”(x) the exponential function exp(fi (A, x ) ) and call elements of C(RN, EN) of the form f(x) = ue’(x), u E E, the plane waves. We use notation
f =1
Uj.e’:(X)
(2.2)
for finite linear combinations of plane waves. For 1 < i < j d N denote by h, the hyperplane {xi = xi} in RN, by sij E W the reflection xi + Xi, xi -+ xi of RN about h, and set RN, = {x1 < . . . < x,}. The following facts are well known. Reflections si,;+ ,, i= l,..., N- 1, generate the group W and RN is the union of “chambers” C,,. = HJRN,, w E W. The rule w -+ C,,. establishes a one-to-one correspondence between chambers and elements of W. Two chambers CL,= URN, and C,,. = WRY are called adjacent if they have a common “wall” hi,. A sequence C I ,..., C,, of chambers is called a gallery if for any k = l,..., n - 1 the chambers C, and C,,, are adjacent. A gallery starting of the “fundamental chamber” C, = RN, and ending at C,,. defines a decomposition w=sj
(2.3)
n j”“‘s,,j,’
and if u’ = 1 (such galleries are ca.lled loops) we obtain the relation SjnL . si, j, = 1.
(2.4)
All relations in W follows from the basic relations which are of 2 types. The first one consists of relations s;.= 1
(2.5)
and the second type involves triples of indices 1
s23s13s12s23s13s12
-
(2.6)
It corresponds to the loop shows on Fig. la. Another way to write (2.6) is s23s13s12
=
s12s13s23.
(2.7)
The sides of Eq. (2.7) correspond, respectively, to the counterclockwise and the clockwise galleries leading from C, to C (see Fig. 1b). All these assertions hold (with slight modifications) for a wider class of groups called reflection groups (cf. [4] or [9]). We use the notation of Section 1 to define the Bethe Ansatz in the general case. So we assume that p 2 1, d> 1 and the matching operators A”(A, p), &,I, ,u) for the two-particle problem are given. Recall that the matching operators (sometimes we
BETHE
ANSATZ
AND
23’13
Xl <
27
EQUATIONS
YANG-BAXTER
si2s23s13s12c+
52
x3
gallery c, to c_ c+ X4<
X2
i.;;i ,‘23
rl
4 Cx3
X3
sl2
FIG.
l(a)
Loop
J
corresponding
to Eq. (2.6).
(b)
two galleries
corresponding
to Eq. (2.7 ).
28
EUGENEGUTKIN
call them matching E,=(@NCP)@Cd.
coefficients)
satisfy equations
(1.3))( 1.6). For N> 2 let
DEFINITION 1. Let N be any integer > 2 and let C(RN, EN) be the space of continuous functions on RN with values in E,. Let A”(& p), B(A, ,u) be a pair of matching operators. A functionfe C(RN, EN) is called a Bethe Ansatz state if the following two conditions are satisfied.
(1) For any w E W the restriction off on C,,. is a linear combination of plane waves. (2) Let C, and C,,. be two adjacent chambers and let h, be the wall between them so that C, corresponds to {xi d *xi> and C,,. corresponds to {xi 3 x,). Let flc, =c
(2.8
u(A) e”(.x),
where n = (2 ,,..., AN). Then flc,, =C [A(A;, Ai) u(A) e”(x) + &A,, A,) u(A) e~+‘(-~)l.
(2.9
Intuitively, a Bethe Ansatz statefis given by a linear combination of plane waves in every chamber and the expressions of f in adjacent chambers match. The problem is to characterize the matching coefficients which allow a full set of Bethe Ansatz states. DEFINITION 2. We say that the matching coefficients A(& p), &%, p) satisfy the full Bethe Ansatz if for any N> 2, almost any 2 = (A, ,..., 2,) E CN and any p E E, there exists a Bethe Ansatz state f such that
f 1c, = ue’(x).
(2.10)
3. We say that matching coefficients A(A, p), B(i, 11) satisfy the (antisymmetric) Bethe Ansatz if for almost any /i = (2, ,..., %N) E CN
DEFINITION
symmetric
p exp(i(fx+my))
A(l,m)p
exp(i(fx+my))+
B(l,m)p
exp(i(mx+ly))
I X
FIG.
2.
Matching
X’Y
coeflicients
for the Bethe
Ansatz.
BETHE ANSATZ AND YANG-BAXTER
EQUATIONS
29
(N 3 2) and any u E E, there exists a Bethe Ansatz state j’which is symmetric (resp. antisymmetric) in X, ,..., -yN and such that (2.1 I ) with ~(1, n)=u.
3. SCATTERING
OF THE TYPE ONE: FULL
BETHE ANSATZ
Recall that p = 1 and the matching operators A(;., p), B(i., p) act on C”. Assume that B(E.,p) (and therefore A(A, ,u)) is a meromorphic matrix function of (k ~0 and that A(& p) + 1 when 3.-p + co (these conditions are satisfied in applications, see Sect. 6). Then det A(L, p) is a meromorphic function which is not identically zero. PROPOSITION
1. Consistenq~conditions for the tnutching oprrtrtors ore ryuiwlen t
to (3.1 )
A ( j., p) + B( 2, 11)= 1 and B(p, 2) = -B(i,
(3.‘)
~1).
Proof: Since in our case A = 2, B = B, Eqs. ( 1.4) and (3.1 ) coincide. Equations (1.5) and (1.6) become B’(E., p) + A(p, 2) A()., p) = 1
(3.3)
A(;., ,u) B(L p) + B(p, j.) A(i. p) = 0.
(3.4)
and
Substituting A = 1 ~ B into (3.3) and (3.4) we obtain B’(I.,~)+(l-B(~,I))(I-B(I,~1))=1 and (1-B(LpO)
B(/1.~)+B(~1,i)(l-B(r.,~o)=O
implying
and -B’(k
P)- B(p, 1) B(L /A)= -B(&
p-
B(p, r,).
30
EUGENEGUTKIN
Hence (3.3) and (3.4) are equivalent. we have from (3.4)
Since B(A, II) commutes with A(1, p), by (3.1),
[BOYPL)+ B(P>A)1 A(& 11)= 0 and multiplying
(3.5)
(3.5) by A ~ ‘(A, p) ( recall that det A(& p) # 0) we obtain (3.2).
PROPOSITION 2. Let A(k p), B(l, 11)be a pair of matching coefficients. The full Bethe Ansatz holds if and only if the functional equations
Ah
v) A(A v) AtA PL)= A(A PL)A(& v) A&, VI,
A(k v) Ah, v) WA, cl) = B(A PL)A(k v) Ah
Ah
(3.6)
VI,
(3.7)
A(& v) A(& P) B(PL, v) = B(P, v) A(4 v) A(k r-l),
(3.8)
2) B(k v) A(& P) + B(A P) BbL, v) B(A PL) = A(v, P) @A, v) Ah, v) + B(P, v) B(A P) HP, v),
(3.9)
A(& PL)WL, v) 41, P) + B(P, 1) B(L v) 42, P) = B(k v) A(A P) B(P, v),
(3.10)
Ah
(3.11)
v) BbL, v) B(i, P) + B(v, P) WI-, v) Ah, v) = WA, v) Alp, v) B(A PL)
are satisfied, which are understood as identities for meromorphic operator functions A(., .) and B(., .). Proof:
Set N= 3 and let f be the Bethe Ansatz state such that flc+ = ,,Ji(1.‘;+u?.+\.I)
Take the counterclockwise gallery leading from C, to C~ = s,~ C, (see Fig. 1). Going along the gallery from a chamber to the adjacent one and using the matching equation (2.9) we determine the coefficients p(w, /1) of the expansion (A = (4 P? v)), (3.12) flc_ =Cu(w, A) e”“(x). II’ Doing the same for the clockwise gallery from C, to C another expansion f 1C_= C u(w, A ) e”‘“(x).
(see Fig. 1) we obtain (3.13)
The formulas for u(w, ,4) and v( w, /i) are obtained by straightforward computations parallel to those of [S, Sect. 4 (we spare the details). We have, for instance, u(l,A) = A@, v) AU, v) 42, pL)u
(3.14)
v(l, A) = A(& p) A(J4 v) A(/& v)u.
(3.15)
and
BETHE ANSATZ AND YANGBAXTER
EQUATIoNS
31
The full Bethe Ansatz holds for N= 3 if and only if for any MIE W and almost all A = (4 11,VI, u(w, A) = u(w, A ),
(3.16)
which gives 6 vector equations. Since u is arbitrary, (3.16) is equivalent to 6 operator equations (3.6)-(3.11). For instance, for w = 1 we have, by (3.14) and (3.15),
which is equivalent to (3.6). For any N> 3 relations in W, are implied by (2.5) and (2.7). Translating this assertion to the language of galleries and loops we see that any loop in RN is a composition of loops conjugate to that on Fig. la. Equations (3.6)-(3.1 I ) mean that the matching is consistent along any such loop, therefore along the composition of any number of such loops. Thus Eqs. (3.6)-(3.11) are equivalent to the consistency of matching along any loop for any N, i.e., to the full Bethe Ansatz. THEOREM 1. Let A(& p), B(1, p) be a pair addition to (3.1) and (3.2) the assumption that
of matching coefficients
satk[\?ng in
det B(j., p) # 0
(3.17)
WA,PL)= [F(n) - F(P)1 ‘,
(3.18)
identical1.v. Then
where F is a meromorphic operator-valued function such that F( E.) commute,fbr all i and det [F(n) - F(p)] is not identically zero. Conversely, if F is any such ,funr.tion then the matching coefficients given by (3.18) satisjj the full Bethe Ansat:. The following lemma, which is needed to prove Theorem 1, seems to be of independent interest. LEMMA 1. Let E be a complex finite-dimensional linear space, and let A(x, J,) he a meromorphic function on C x C with values in the space M(E) of operators on E. Assume that A(x, y) and A(z, u) commute if there are no more than 3 d&inct numbers in {x, y, z, u}. Then A(x, y) and A(=, u) commute for all X, J, :, and u.
Postponing the proof of Lemma 1 we now prove the theorem. Proof of Theorem 1. First we will show that Eqs. (3.6)-(3.8) are equivalent to the commutativity of A(;1, p) for all A, p. Equation (3.7) means that B(& 11) commutes with the product A(i, v) A(p, V)
32
EUGENE
GUTKIN
therefore, by (3.1), A(,I, p) commutes with A(& v) A(p, v) for any A, p, and v. From this and (3.6) we have
which we rewrite as
Cd& VI,44 VII A(& PL)= 0. Multiplying (3.19) by A ~~‘(II, p) we conclude that satisfies assumptions of Lemma 1. Therefore all satisfied. Conversely, if {A(i, p)} is a commutative obviously satisfied. Now the order of factors in (3.9))(3.11) does rewrite (3.10) as
(3.19)
the family {A( , )) of operators A(& p) commute and (3.8) is family then Eqs. (3.6k(3.8) are not matter and, using (3.2), we
44 PL)CHAPL)m v)-B(A PI HA v)-B(A VI m VII =a Multiplying
(3.20)
(3.20) by A(1,p) ~ ’ we get
CNA 11)+ m.4 v)l B(J.,v) = WA PL)m, v).
(3.21)
Dividing (3.11) by A(p, v) we also get (3.21), hence (3.10) and (3.11) are equivalent to each other and to (3.21). Rewrite (3.9) as
[A(& P) Ah A) - A(K v) A(v, ~11WA VI = B(A ~1 WL, v)CWp,v) - W, pIl(3.22) and use (3.1) to get rid of A in (3.22). After some simplifications
we obtain
[m4 vJ2- NJ”?PI’1 B(A v) = WA,PI m, VKm4 v) - B(4 PL)l and factoring Q,
v) - B(I, p) out we have
CQL, v) - B(k PL)IC(B(LP) + WP, v)) @A v) - WA ~1%A ~11=O, (3.23) which of course follows from (3.21). Thus the system of equations (3.6)-(3.11) is equivalent to the commutativity of matching coefficients and Eq. (3.21). Denote B(& CL))’ by G(& p) and multiplicity (3.21) by G(& p) G(p, v) G(& v). We obtain
(31, PL)+ G(P, v) = (32, v).
(3.24)
To work with this equation it is convenient to denote ;1, p, v, by x, y, z, respectively. Using (3.2) we rewrite (3.24) as G(x, z) = G(x, Y) - G(z, Y).
(3.25)
BETHE
ANSATZ
AND
YANG-BAXTER
33
EQUATIONS
Since &G(.K.
.I’): =
lim G(x+A, 1- 0
y+ A)-G(.u+A,
.v-G(-u, A2
y+A)+Gk
.v)
(3.25) implies that (3.76) The general solution
of (3.26) is G(x, y)=g(-~)+h(g)
and (3.25) implies that h(.~) = -g(s).
Denoting
g by F we have
G(.Y, y) = F(s) - F(y),
(3.17)
which is equivalent to (3.18). Of course, the function F(s) is defined by (3.27) only up to a constant operator. Since G(x, ~1) commute with each other, we can choose this constant so that the operators F(s) also commute. For any operator function F(x) satisfying the assumptions of Theorem 1 the function G(s. J,) given by (3.27) satisfies (3.24). therefore B( X, y) = G(.Y, J,) ’ satisfies (3.21) which implies (3.6)-( 3. I 1). This proves the theorem. Pm$c?f’ Lumm 1. Since the function A(x, J.) is meromorphic of points (s, J’), where A(s, 1’) is defined, is open and dense in the exposition we will argue as if R = C’“. A subspace F c E is an eigenspace of a set X of operators on there is a number i. (eigenvalue of .4 on F) such that (A - l)“F= any commutative family X of operators there is a unique direct E=xE, !El of E into eigenspaces of X such that if E,,(A) is the eigenvaiue ).,(A ) # i,(A) for at least one A E X. Fix .YE C and let E=x
E,(.Y)
on C’, the set R C’, and to simplify E if for any .A E ,I 0 for some H. For decomposition (3.28) of A on E, then
(3.29)
be this decomposition for the commutative family (A(.Y, ~2): J-E C)-. We denote by E.,(s, J,) the eigenvalue of ,4(x, ~1) on E,(x). For generic y we have j-,(.x, J) # j*,(.u, y). therefore for generic J’ the eigenspace E,(x) can only split if we move horizontally away from (.I-, y). Since dim E,(.K) 3 1 we conclude that there is a unique decomposition (3.28) of E
34
EUGENEGUTKIN
into eigenspaces of the set { A(x, y): x, y E C} and that for almost all (x, v) (call them regular) we have &(x, y) # Aj(x, y). Since now the assertion of the Lemma reduces to the same question in each E;, we can assume without loss of generality that 111= 1, i.e., A(x, y) = 1(x, y) + N(x, y), where N(x, y) are nilpotent and have the same commutativity properties as A(x, y). Subtracting J(x, y) off we reduce the question to the case when A(x, v) are nilpotent. By the same argument as above we show that E contains a nonzero subspace E, such that A(x, y) E, = 0 and E, = Ker ,4(x, y) for regular (x, y). We can now argue by induction on dim E and assume that the induced operators A r(x, y) on E, = E/E, commute with each other. Then the operators ,4(x, -v) obviously commute. The lemma is proved. Remark. Let us drop assumption (3.17) in Theorem 1. The argument of Lemma 1 shows that Cd= E,@ E,, where E, and E, are invariant under B(& p), the restriction B,(& p) = B(1, p)IE, satisfies (3.17) and B,(il, p)=B(& p)IE, is nilpotent for all 1 and CL.Equation (3.21) has plenty of nilpotent solutions. For instance, let dim E, = 2n and let &(,I, p) have the block structure
where C(A, p) is an arbitrary meromorphic function with valus in M(C”). (3.21) is satisfied. But if we require that B,(I1, cc)= 0 then Theorem 1 yields
Then
COROLLARY 1. Let A(& p), B(& ,a) be a pair of matching coefficients. Let Cd= E, @E, be the canonical decomposition guaranteed by Lemma 1 such that det[B(& p)lEOl =0 and det[B(l, p)lE,] #O. Assume that B(l>, p)JE0= 0. Then there exists a meromorphic function F on C with values in M(E,) such that F(n) all commute, det [F(A) - F(p)] # 0 and
COROLLARY 2. Let the matching coefficients depend on I-p, i.e., 44 PL)= A(A - ~1, B(A PL)= B(J. -PI and assumethat B,(13.- ,a) = 0. Then the full Bethe Ansatz is satisfied if and only if B has the form
B(1) = CA - ‘,
(3.30)
where C is an operator on Cd (E, = Ker C). ProoJ: Assume for simplicity
is trivial).
Then, by Theorem
of exposition
that E, = 0 (the extension to E, # 0
1, B-‘(l-p)=
F(%-
F(p).
BETHE ANSATZ AND YANG-BAXTEREQUATIONS
35
Setting p = 0 in the equation we have F(i)=
B-‘(I-)
hence F(I - p) = F(R) - F(p). Besides, by (3.21, B-‘( -2) = -B-‘(I.),
thus F(A) satisfies (3.31)
F(i.+p)=F(l.)fF(p). The general solution of this equation is F(i) = F. i,
where F is an arbitrary nondegenerate matrix. Setting C = F ’ we obtain (3.30). COROLLARY
3. Let a(],, p) and h(A, p) he a pair qf’scalar matching coeJficiPnts.
i.e., d= 1. (i )
The)! sati$v the fidl Bethe Ansatz tf and or&j if’
where g is anI> nonconstant meromorphic jknction. (ii) ff a and h depend on I. - ,u on!,+, then the ,full Bethe Ansut: is sati$kd if and only ij h(i,,u))=C(~-11)
‘.
(3.33)
bcherec is an arbitrary complex number. Proof: Assertion (i) follows from Corollary 1 since its assumption is trivially satisfied for d = 1. Assertion (ii) obviously follows from Corollary 2.
4. SCATTERING SYMMETRIC
OF TYPE ONE: AND ANTISYMMETRIC BETHE ANSATZ
Recall that p = 1, E, = C”, the action of W, on E,,,, is trivial and (2.1 ) becomes
THEOREM 2. (a) Any pair A(,l, ,D), B(,l,p) antisymmetric Bethe Ansatz.
oJ1 matching coefficients satisfks
(b) Let A(R, p), B(,l, p) he a pair qf commuting matching co@cients. Then the symmetric Bethe Ansat; is satisfied.
36
EUGENE
GUTKIN
Proof It is convenient to change notation Bethe Ansatz state fin a chamber C,. as
slightly and write the expansion of a
flc, =I 4~ A) e”(x),
(4.1)
A
where the summation We have
is over all /i belonging to a W-orbit.
Denote u( 1, /i) by u(A).
flC,(4=flC+w’x)
(4.2)
flc,(X)=(-)‘~lc+(~~~‘~~)
(4.3)
and
in the symmetric and antisymmetric case, respectively (where (- )“‘ means det w). From (4.1), (4.2), and (4.3) we immediately obtain u(w, A) = u(u’-‘A)
(4.4)
u(w, A) = (- )“‘u(w’A)
(4.5)
and
in the symmetric and antisymmetric cases, respectively. Let C,. and C, be two adjacent chambers and let s = sij be the reflection about the wall between them where C,,. corresponds to x, < xi. Consider the coefficients u(w, A), u(u’, sn), u(u, A), and u(u, sn), where n = (A,,..., AN). By (2.9), (4.4), and (4.5) we have u(u, A)=
fu(w, s/l),
u(u, s/l) = +qw,
(4.6)
A),
(4.7)
u(u, A) = A(l*;, /Ii) U(W, A) + B(A,, ni) u(w, s/i),
(4.8)
u(u, s/l) = A(A,, Ai) U(W, s/f) + B(&, II,) u(w, A),
(4.9)
where the plus sign (resp. minus sign) corresponds to the symmetric (resp. antisymmetric) case. Solving (4.6) for u(w, sn) and substituting into (4.8) we obtain after elementary transformations [l
fB(Aj,
where now minus corresponds obtain from (4.10)
ni)]
U(U,
A)=A(A,,
A,)
to the symmetric
A(A,, ;1,) u(“, A)=A(ni,
Aj)
U(Wy
A),
(4.10)
case. Using (3.1) and (3.2) we u(w, n)
(4.11)
BETHE
ANSATZ
AND
YANCBAXTER
37
EQUATIONS
and (4.12)
A(&, A,) u(u, A) = A(E.,, R,) zf(M’, A)
in the symmetric and antisymmetric (4.11) and (4.12) are equivalent to
cases, respectively. Since A(& ,u) is invertible,
II(L), A) = A ‘(A,, A,) A(/?,, I.,) tf(w, A)
(4.13)
u(v, A) = U(M., A).
(4.14)
and
respectively. NOW solve (4.6) and (4.7) for U(U, /1) and U(L),sA), respectively, and substitute into (4.8) and (4.9). We obtain [k 1 - B(i,, ii)] u(M~,sA)=A().,,
(4.15)
i,)u(ti’, A)
and [ fl
- B(A,, A,)] u(M’, A)=A(l.,,
which are equivalent since /i is arbitrary From (4.15) we get
i,) u(tt‘, .x/l),
(plus corresponds to the symmetric case),
u(M’,sA)=A-‘(~~,~.~)A(~“,,~.,)u(I~.,
(4.16)
A)
and (4.17)
u( h(‘, s/i) = -u( M’, A )
in the symmetric and antisymmetric cases, respectively. The Bethe Ansatz statefcorresponding to u E C” and A, E CN exists if and only if the system (4.6)(4.9) of homogeneous linear equations (where M’, u E W and II = g/i,. gg W) has a solution u(g, A) with U( 1, A,) = U. Equations (4.6).-(4.9) are equivalent to (4.13) and (4.16) in the symmetric case and to (4.14) and (4.17) in the antisymmetric case. The obvious solution of the latter system is u( g, wAo) = ( - )“‘u,
(4.18)
which settles the antisymmetric Bethe Ansatz. If A,, is regular, i.e., IVY,= A, implies HZ= I, then Eqs. (4.13) and (4.16) are equivalent. To show the existence of Bethe Ansatz states it is convenient to work with the system (4.13). This system has a solution if and only if for any relation (2.4) in W we have (/iO = (J ,,..., A,)) A ‘(i.,,l.in)A(i,n,
n,~)...A-‘(~,,,i.,,)A(l.,,,
i,,)=
I.
(4.19)
38
EUGENEGUTKIN
It suffices to check (4.19) for the basic relations (2.5) and (2.6). Equations (4.19) corresponding to relations (2.5) are trivially satisfied. Replacing (2.6) by (2.7) we rewrite the corresponding equation as
which is obviously theorem is proved.
satisfied if the matching
coefficients A(5 ,u) commute.
The
COROLLARY 4 (of the proof). Let A(& p), B(I, p) be any pair of matching coefficients. They satisfy the symmetric Bethe Ansatz if and only if Eq. (4.20) holds.
Remark. The argument of Theorem 2 also answers the following question. We say that the weak form of Bethe Ansatz holds if for almost all A = (A, ,..., AN) there is a u # 0 which determines the symmetric Bethe Ansatz state (2.11). Denote by A(IZ,, 1,, &) the left-hand side of (4.19) corresponding to the relation (2.6). COROLLARY 5. Matching coefficients A(& ,u), B(1, ,u) satisfy the weak form of symmetric Bethe Ansatz if and only if the equation
det[A(L, CL,v) - l] = 0
(4.21)
holds. ProoJ: By the argument of Theorem 2, the weak form of symmetric Bethe Ansatz holds if and only if for almost all (A, ~1,v) there exists u # 0 such that A(n, ,u, v)u = u which is equivalent to (4.21). We will calculate the expansion (2.11) of symmetric Bethe Ansatz states. First introduce some notation. A gallery C, = Co, C, ,.,., C, = C, leading from C, to C, defines the sequence (iI, jI) ,..., (i,, j,) of orde re d pairs of indices given by the rule that C,_ i is contained in the halfspace (xi, < xj,}. The ordered pairs (i, j) correspond to the roots y of the symmetric group W (cf. [4]). We denote the situation described above by u-v Y,s...,Yn
(4.22)
and call y1 ,..., Y,, the gallery sequence of roots. If y = (i, j), denote the reflection sii by s,, and set -y = (j, i). If y = (i, j) and A = (2 ,,..., A,,,), denote &- lj by (y, A) and set A(Y, A) =A(ni, Aj). The roots y = (i, j) with i< j are called positive (y > 0) and the roots ai= (i, i+ l), i= l,..., N - 1, are called simple. Reflections 0; = s~,~+i, i = l,..., N- 1, are called simple. A decomposition W=o,“‘cT”
(4.23)
BETHE
ANSATZ
AND
YANG-BAXTER
39
EQUATIONS
of WE W into a product of simple reflections is called reduced it it has minimal length. Denote by y + wy the natural action of W on the roots. Let (4.23) be any decomposition and let a,,..., c(, be the corresponding simple roots. Consider the sequence u,, CJ~~~-, ,..., (T,. . . c, = IV-’ of group elements and the sequence y, = a,,, y2 = ~,,a, ~, ,..., yn = (o,, . az) 01~of roots. Then 1-w
I
(4.24)
y, . Yn
and (4.23) is reduced if and only if y ,,..., yn>O. The roots y=y ,,..., y,, in (4.24) depend only on w and they are characterized by y > 0, MT?< 0 (i.e., - M’Y> 0). The assertions above hold for general reflection groups and the reader can find proofs in [4] or in [7,9]. THEOREM 3. Let A(A, p), B(l., p) be commuting matching operators and ,fin u E Cd, A E CN denote by f,,,* the symmetric Bethe Ansat: state which has the e.upansion
fu.,,IcL =Cu(wA)e”“(x)
(4.25)
ulith u(A) = u (by Theorem 2, f,,,, exists ,for all II and almost all A ). Then u(wA)= [I-I.ico A(;‘, A)]-‘[n,,, A(y, wA)lu, i.e., (4.26) ProoJ: The pairing
(y, A) is invariant
with respect to the action of W:
( y, A ) = (WY, tt,/l ).
(4.27)
Let (4.23) be a reduced decomposition and let (4.24) be the corresponding gallery sequence. Set u,, = g,,, U, ~, = o,,cn , ,..., U, = CJ,, (T, = w ‘. The chambers UsC’ +. C + are adjacent for k = O,..., n - 1. and uk+l Let z be a simple root, let s be the corresponding reflection and let VE W. From (4.16) setting u = 1 and replacing n by VA we have zc(svA)=A~~‘(-r,vA)A(a,vA)u(vA)=A
Applying steps
‘(-0
(4.28) successively to 0,~’ ‘..o,,A, u(wA)=A
‘(-u,M,,
A) A(uzul, A)...
‘cr,A)A(v
oza,...~,z~
‘x,A)u(vA).
(4.28 )
,..., o,,A we obtain after II
A ‘(--a,,. A) A(cz,,, A) u(A).
(4.29)
The roots ?,I = a,,..., y, = uzcyl form a gallery sequence (4.24) and, since (4.23) is
40
EUGENEGUTKIN
reduced, they are characterized by y>O, wy ~0. Using that A( -y, A)= A(y, -A), that A(, ) commute and that u(A) = u we rewrite (4.29) as u(wA)=
n
A-‘(y,
-A) A(y, A)
u.
(4.30)
I
YSO
WY<0 Set A(w,A)=
(4.31)
n A(y,A). y>o ny
Then (4.30) can be written as fu,/,Ic.+ =I
II’
A(w, A) A-‘(w,
-A)
u&““(x).
Let wo: 1 --) N, 2 + N - l,..., N -+ 1 be the longest element multiply (4.32) by A(wo,
-A)=
(4.32) of W and let us
n A(y, -A). YSO
(4.33)
We have for any M’E W, A(w, A) A-‘(w,
-A) A(w,, -A) (4.34)
= n A(Y, A) n NY, -A). i’>o v>o wg -c0 ““i > 0 Using that A(y, -A) as
= A( -y, A) and (4.27) we rewrite the right-hand
side of (4.34)
fl A(Y, -WA), y>o which means that
u(wA)= n Ah,-wA)lA(y, -A)= n A(Y,wA)lA(y,A) y>o
(4.35)
Y
and proves the theorem. COROLLARY 6 (of the proof). Let A(& p), B(2, 11) be commuting matching coef ficients. The symmetric Bethe Ansatz state fu.n exists for A = (A,,..., I,) if the operators A(n,, Azi)are definedfor all i # j and det A(Ibi, A,) # 0 for i < j. It is giuen by
fu,~lc+Ew~w[
n A(y,/f)lA(y, -~)]~~Y.~).
Y>O M.y
(4.36)
BETHE
ANSATZ
AND
YANG-BAXTER
41
EQIJATIONS
Proof: Formula (4.36) follows immediately from (4.30) obtained in the proof of Theorem 3. It makes sense if the operators A(y, /i) are defined for all 7, that is A(L,, I.,) exist for i #j, and the operators A(y, -,4) are invertible for 1’ > 0, that is, det A(L,, L,)#O for i
n A(i,,,. I(,,, E.,,.-+,,)/A(jU,. i.,)
I(,
A().,, ).,)/A(;.,.
II
;,)
ue” ‘(.Y)
(4.37)
ue”.‘(.‘i),
(4.3X I
1
‘(i)
'(il>bt
1
Lchere the summation is over all perrmtations 11’of 1l..... N) (b) Let the nlatching coefficients A()., p). B( A, p) he given bus B(j,, p) = C(3. -p)
(4.39)
‘.
where CE M(C”). Then the s~.wmetric Bethe Amutz stute f;,. , e.xisf.s fiw uimo.vt till A = (2, ,..., &,,) and is given by the equivalent e.upre.s.sion.s
und .Ll
c+ = C det 11’
,I’
n
I
c-(&-a,, C-(i,-2,)
ue”‘,‘(s).
(4.41 )
Proof Formulas (4.37) and (4.38) are straightforward translations into the language of permutations and ordered pairs (i, .j) of (4.26) and (4.36), respectively. Formulas (4.40) and (4.41) are derived from (4.37) and (4.38), respectively, by substituting A(k
p) = [(1.-p)
- C]/(l. - p).
42
EUGENEGUTKIN
COROLLARY 7. Let A(& u), B(I, p) satisfy the full Bethe Ansatz and depend on I- p. Then the symmetric Bethe Ansatz state f,,n exists if lj - Izi is not an eigenvalue of C for all i< j and it is given by (4.40) or by (4.41). Proof By Corollary 2, B(& p) has the form (4.39) therefore fu,n exists for almost all /i and is given by (4.40) or, equivalently, by (4.41) if the formulas make sense, which they certainly do if det( C- (lj - A,)) # 0 for i < j. 5. FULL BETHE ANSATZ AND GENERALIZED YANG-BAXTER
EQUATIONS
In this section we set p > 1, d = 1, and study the case (ii) of Section 1, i.e., scattering between particles of different kinds. We recall from Section 1 that the matching operators are T(2, p) and R(I, p) and they act on Cp@ Cp = Cp2. They satisfy the consistency conditions T(4 CL)+ R(A P) = 1,
(5.1)
R*(k P) + Tbu, A) T(A P) = 1,
(5.2)
and T(L P) N4
P) + NCL, 1) W, 11)= 0.
(5.3)
As in Section 3 we assume that the matching coefficients depend meromorphically on Iz and p. We no longer assume that A(& p) + 1 when i -CL + co since this is not so in many applications. The cases T(& p) = 0 or R(A, p) = 0 identically are trivial and we exclude them in what follows. Proposition 1 in the present context becomes PROPOSITION 3.
Assume that det T(2, p) # 0 identically.
Then
N/4 A) = -Nil, P).
(5.4)
The space E, of Section 2 is equal in this case to ON Cp and W, acts on ON Cp by permutations of factors. Now Definitions 1, 2, and 3 of Section 2 apply in the present case. Let A be an operator on Cp 0 Cp. For 1 < i < j d 3 we denote by A, the operator on Cp@ Cp 0 Cp which is given by A on the product of the factors i and j in the tensor product and is equal to identity on the remaining factor. For any 1 < i < j< 3 the correspondence A + A, is an isomorphism of M(CP@ Cp) into M(CP@ CP@ cq. THEOREM 5. Let T(2, p) and R(A, p) be a pair of matching operators for the twoparticle scattering (p > 1). Then the full Bethe Ansatz holds if and only tf the matching operators satisfy the equations T&t T23(4
v) T,,(A v) v) T&u,
v)
T,2(4 R,2(4
v) =
T,,(k
PU)T,,(k
v)
T23(4
v),
(5.5)
P) =
R12(4
PL) T,,(L
v)
T23bL,
~1,
(5.6)
43
BETHE ANSATZ AND YANGBAXTEREQUATIONS
= R,,(v, P) R,,(A v) TAP, v) + T&L, v) R,,(k P) MP,
(5.8)
~1,
R,,(P, 1-1Rn(A v) T,,(A P) + T,,(A cl) R,,(P> v) R,z(L PL) = R,,(k v) T,,(k PL)R&L, v), T,,(P~ 4 R,AL v) T,AA PL)+ RAA
(5.9) PL)R,,b,
= T,,(v, cl) R,,(A v) TAP, v) + R&L
v) Rdj., P)
v) R,,(k PI R&,
VI.
(5.10)
Proof Equations (5.5)-(5.10) are obtained by matching the Bethe Ansatz states for three scattering particles. They are the analogs of Eqs. (3.6)-(3.11) in the present situation and the proof of Theorem 5 is completely analogous to the proof of Proposition 2. We spare the details. COROLLARY 8. Let the matching coefficients To., p) and R(,?., p) depend on the difference 1. - p. Then the full Bethe Ansatz is satisfied if and only if the equation.7
T,,(u) R,du + uV’,,(u) = T,,(u) T,,(u + u) T,,(u),
(5.11 )
Tzdu + u) T,&u) R,Au) = R,Au) T,,(u + 1,) TJu),
(5.12)
R&u) T,,(u + u) T,,(u) = T,,(u + t’) T,,(u) RX(U).
(5.13)
R,,(u + u) T,,(u) R,,(u) = Rd-0)
R,,(u) T,,(v) + T,,(u) R,,(u) Rzdo),
R,,( -u) R,,(u + u) T,,(u) + TAu)
(5.14)
R,,(u) R,z(u) (5.15)
= R,,(u + 0) Tutu) R,,(u), Tzd -u) R,,(u + u) T,,(u) + Rn(u) R,,(u) R,z(u) = T,,( -u) R,,(u+ 0) T,,(u) + R,z(t;) R,,(u) R,,(u)
(5.16)
hold. Proof. Equations (5.1 lt(5.16) are obtained from (5.5)-(5.10) by substitution A-p=u, p-v=u, E,-v=u+u. Equation (5.11) for the operator function T(u) is known in the literature as the Yang-Baxter equation. Recently it came up independently in the work on factorizable S-matrices [ 193 and on the soluble models of statistical mechanics [ 11. In view of this we propose to call the system of equations (5.5)-( 5.10) or (5.11)-( 5.16 ) the generalized Yang-Baxter equations. The following is immediate. COROLLARY
9.
Let
the matching
operators
T(A, p), R(A, ,u) depend
on A-p.
[f’
44
EUGENEGUTKIN
the many-particle scattering problem has the fill Yang-Baxter equation (5.11).
Bethe Ansatz then T(u) satisfies the
6. EXAMPLES Case (i). We will consider singular differential operators on P-valued on RN of the form G = -d + singular terms
functions (6.1)
where A= 2
a2/ax;
(6.2)
i=l
is the standard Laplacian and the singular terms are delta-function-type singularities supported on the hyperplanes {xi = xi). An operator (6.1) is the positive Laplacian with boundary conditions on {xi = xi, i, j = l,..., N} determined by the singular terms in (6.1) (see examples below). These boundary conditions define a pair A(& p), B(L, p) of matching coefficients satisfying the conditions (3.1) and (3.2). The corresponding Bethe Ansatz states (see Definition 1) are called the Bethe Ansatz eigenstates of G. They are in fact the generalized eigenfunctions of G since they satisfy -Af=Ef (6.3) and the boundary conditions imposed by G. We say that the operator (6.1) satisfies the full (resp. symmetric) Bethe Ansatz if the corresponding matching coefficients satisfy it (see Definitions 2 and 3). EXAMPLE
1 (Delta-potential).
Let C be any d x d matrix and set
H= -A+2C
1 &xi-xi).
(6.4)
i-zj
Denote by a/ax, - a/ax,)fl+ the jump of the “normal derivative” (a/axj - a/ax,)f on the hyperplane (xi = xi}. It is standard to check that the boundary conditions imposed by (6.4) are - a/ax,)fI f = 2cf
(6.5)
for l
to (6.5) consider the case of
(a/axi
(6.6)
BETHEANSATZ
AND YANG-BAXTER
45
EQUATIONS
The left-hand side of (6.5) is the restriction where i= fi. i(~ _ ~) Uel(;..,+ll.,‘)+i(j”-~)A(~,~)uei(‘-~+~““+i(~-i)B(~,IOue”~”+”’
to r = .Y of which
gives, taking (3.1) into account. 2i(p - 2) B(A, p) ue”L+l”r.
(6.7)
The right-hand side of (6.5) is 2C’ue”” + I”’ hence B(/l,~)=C/i(~c-it)=iC(j.--~)
‘.
(6.8 1
Thus the operator (6.4) satisfies the full Bethe Ansatz for any matrix C. Hence the symmetric Bethe Ansatz is satisfied and denoting by ,/;,, , the symmetric Bethe Ansatz eigenstate of H corresponding to II E C” and iI = (E,,..... E,L ) we have from (4.40)
with the summation over all permutations M’of i l...., IV;. Operator (6.4) is selfadjoint if C* = C and in the scalar case ti= 1, C is a real number, H is the famous Hamiltonian with delta potential (cf. [2, 14, 13, 18, 61). EXAMPLE
2 (Delta shift).
Let C be any clx ti matrix and set
L = A + 2c 1 6(x; - .u,)(?/?s, + C/?-Y,1. I< ,
(6.10)
The boundary conditions imposed by (6.10) are (d/Ss, - (‘/&Y,,fl * = -2C( ?/?s, + ?/?.u,),f:
(6.11 )
Computing the matching coefficients as in Example I we get the equation
which implies
By Corollary 2 to Theorem 1, the full Bethe Ansatz is not satisfied unless C = 0. However, by Theorem 2(b), L satisfies the symmetric Bethe Ansatz for all C. From (6.12) we have, by (3.1),
46
EUGENEGUTKIN
Substituting (6.13) into (4.37) we obtain a formula for the symmetric Bethe Ansatz eigenstate fu,n of L. The product in (4.37) becomes (we use summation over w instead of W-‘)
n [(n,(j)-n,(j))-(n,(j)+n,(i))C](S-ni)
icj
C(~j-ni)-(~j+ni)c](n,,j,-n,,i,)
=(-)“’
(‘w(j)
n
-‘W(i))
-
(l,(j)
+
n,(i))C
(6.14)
(A,j-Ai)-(lLj+Izi)C
i-cj
’
Using that
II (‘W*(i)+ ‘w,(j))= n (Ai + Aj), icj
i
we see that (6.14) is equal to
(-y& +n.,(j)-C]y[g-c]~ A,(j)
-
in(i)
I
41)
J
therefore
x explfi
(L(,,xl + ... +&v~X~)I.
(6.15)
In the scalar case d= 1 and C is a real number, L is the generating operator of the semigroup describing the motion of N Brownian particles with delta-shift. This operator is connected with the Burgers’ equation [ 151 and the Bethe Ansatz for L was used in [ 10, 1 l] to linearize the equation. EXAMPLE
arbitrary
d
3 (General delta-interactions). Fix x d matrices. Consider the operator
G= -d+2
n 30
and
let
Co,..., C,
be
C d(Xi-Xj) i
X [Co-C,(d/dX;+d/dXj)+
The boundary conditions (a/axi -
a/aXj)fl
’ ’ ’
+
( -
)“C,(a/CTXj
+
a/aXj)“].
(6.16)
imposed by G are +
=2[C,-C,(a/axi+a/axj)+
. . . + ( - )“C,( d/3x, + d/dx,)“]f:
(6.17)
BETHE ANSATZ AND YANG-BAXTER
Computing
the matching
coefficients as before, we get the equation
2i(p-i)B(I,p)=Z[C,--i(R+p)C,+
47
EQUATlONS
(i = \/ - 1 ) (6.18)
... +(-i)“(i+p)“C,,],
from which we have B(j.,~f)=[iC,+(;!+~)C,+
... +(-i)“~‘(3.+~)“C,,](E.-~O
‘.
(6.19)
In view of Theorem 1, G satisfies the full Bethe Ansatz if and only if C, = ... = C,, = 0, thus delta-potential is the only delta-interaction satisfying the full Bethe Ansatz. If the operators Co,..., C, commute, formula (6.19) shows that all B(E,, ,LI) commute, therefore, by Theorem 2(b), G satisfies the symmetric Bethe Ansatz. Assume that C,,,..., C,, commute and set y(.u)=iC,+sC,
+ “. +(-i)”
‘X’T,,.
(6.20)
Then from (6.19). by (3.1), we have A(i,p)= and, by computations
[(3.-~)-4(E.+~)]:(E.-~f),
(6.11 )
parallel to those of Example 2, we have
x exp[JZ
(&,.,,,.u, +
+ &,.,.,.y,,,)].
(6.22)
Case (ii). Examples of solutions of the Yang-Baxter equation are well known (cf. [12, 191) although the general solutions seems to be out of reach. We will return to the study of solutions of the generalized Yang-Baxter equations (5.5)-(5.10) and (S.ll )-(5.16) in a future publication.
REFERENCES I. R. J. BAXTER, “Exactly Solved Models in Statistical Mechanics. Academic Press. New York, 19X2. 2. F. A. BEKEZIN. G. P. POHIL, AND V. M. FINKELBERG,Schroedinger equation for the system of oncdimensional particles with point interaction. Vestnik Moskor. Unio. 1 (1964), 21. 3. H. BETHE. Zur Theorie der Metalle, 1. Eigenwerte und Eigenfunktionen der Atomkette, Z. P1zy.v. 71 (1931). 205. 4. N. BOURBAKI, “Groups et algtbres de Lie.” Chap. 4-6. Hermann, Paris. 1968. 5. M. GAUDIN. “La fonction d’onde de Bethe,” Masson, Paris, 1983. 6. E. GUTKIN, Integrable systems with delta-potential, Duke Murh. J. 49 (1982). 1. 7. E. GUTKIN, Operator calculi associated with reflection groups, Duke Marh. J. 54 (1987). I-18. 8. E. GUTKIN. Integrable many-body problems and functional equations, J. Marh. And. Appl.. in press. 9. E. GUTKIN, Geometry and combinatorics of groups generated by reflections, Enwigtz. Muh. 32 (1986). 95.
5Y5’176. I -‘I
48
EUGENE GUTKIN
10. E. GUTKIN AND M. KAC, Propagation of chaos and the Burgers equation, SIAM J. Appl. Math. 43 (1983), 971. 11. E. GUTKIN, Propogation of chaos and the Hopf-Cole transformation, Aduun. Appl. Math. 6 (1985), 413. 12. P. KULISH AND E. K. SKLYANIN. Solutions of the Yang-Baxter equation, J. Soviet Marh. 19 (1982). 13. E. LIEB AND W. LIENIGER, Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Whys.Rev. 130 (1963), 1605. 14. J. B. MCGUIRE, Study of exactly soluble one-dimensional N-body problems, J. Math. Phys. 5 (1964), 622. 15. H. P. MCKEAN, JR., Propagation of chaos for a class of nonlinear parabolic equations, in “Math. Studies,” Vol. 19, pp. 1777194, Van Nostrand, Princeton, NJ. 16. B. SUTHERLAND, An introduction to the Bethe Ansatz, in “Proceedings of the Winter School on Exactly Solvable Problems, Panchgani, India, 1985,” pp. l-96, Lecture Notes in Physics, Vol. 242, Springer-Verlag, Berlin. 17. L. A. TAKHTAJAN, Introduction to algebraic Bethe Ansatz, in “Lecture Notes in Physics,” Vol. 242, 175-220, Springer-Verlag, Berlin. 18. C. N. YANG, Some exact results for the many-body problem in one dimension with repulsive deltafunction interaction, Phys. Rev. Letr. 19 (1967), 1312. 19. A. B. ZAMOLODCHIKOV AND AL. B. ZAMOLODCHIKOV, Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models, Ann. Phys. 120 (1979) 253.