Table Of ContentSymmetry, Integrability and Geometry: Methods and Applications SIGMA 3(2007), 013, 14 pages
Relativistic Toda Chain with Boundary Interaction
⋆
at Root of Unity
Nikolai IORGOV †, Vladimir ROUBTSOV ‡§, Vitaly SHADURA † and Yuri TYKHYY †
† Bogolyubov Institute for Theoretical Physics, 14b Metrolohichna Str., Kyiv, 03143 Ukraine
E-mail: [email protected], [email protected], [email protected]
‡ LAREMA, D´ept. de Math. Universit´e d’Angers, 2 bd. Lavoisier, 49045, Angers, France
E-mail: [email protected]
7
0 § ITEP, Moscow, 25 B. Cheremushkinskaja, 117259, Moscow, Russia
0
Received November 15, 2006, in final form January 03, 2007; Published online January 19, 2007
2
Original article is available at http://www.emis.de/journals/SIGMA/2007/013/
n
a
J Abstract. We apply the Separationof Variables method to obtain eigenvectorsof commu-
9 ting Hamiltonians in the quantum relativistic Toda chain at a root of unity with boundary
1
interaction.
] Keywords: quantumintegrablemodelwithboundaryinteraction;quantumrelativisticToda
I
S chain
.
n 2000 Mathematics Subject Classification: 81R12; 81R50
i
l
n
[ To the memory of Vadim Kuznetsov
1
v
0 1 Introduction
4
0
During the last decade, a considerable progress in application of the Separation of Variables
1
0 method to a broad class of quantum integrable models has been achieved. This progress was
7
initiated by the paper [1] of Sklyanin who has proposed (using the R-matrix formalism) a recipe
0
/ for a Separation of Variables in the case of quantum Toda chain where the algebraic Bethe
n
ansatz fails. The next important step was done by Kharchev and Lebedev [2] who realized an
i
l
n iterative procedure of obtaining the eigenfunctions of the n-particle open Toda chain by some
: integral transformation from the eigenfunctions of the (n 1)-particle open Toda chain. This
v
−
i iterative method was applied later to a relativistic Toda chain [3].
X
The Separation of Variables was found to be effective for the derivation of the eigenfunctions
r
a of the commuting Hamiltonians of the n-particle quantum Toda chain with boundary interac-
tion [4, 5] in the framework of Sklyanin’s approach to the integrable boundary problems [6].
In the paper [5] the n-particle eigenfunctions of the quantum Toda chain when the first and
last particles are exponentially interacting with the walls (the two-boundary interaction) is con-
structed by means of an integral transformation of the eigenfunctions for the Toda chain with
one-boundaryinteraction(theauxiliaryproblem). Theseeigenfunctions,inturn,areconstructed
using the eigenfunctions of the n-particle open Toda chain.
Recently, a Separation of Variables was applied to the N-state spin lattice model – the
Baxter–Bazhanov–Stroganov (BBS) model [7, 8]. In the R-matrix formulation of this model the
cyclic L-operator [9,10]emerged. Theseoperators areintertwined by thesix-vertex R-matrix at
arootofunity. ThecyclicL-operatoroftheBBSmodelreduces(atspecialvaluesofparameters)
to the L-operator of the quantum Relativistic Toda Chain (RTC) at a root of unity [11].
⋆ThispaperisacontributiontotheVadimKuznetsovMemorialIssue“IntegrableSystemsandRelatedTopics”.
The full collection is available at http://www.emis.de/journals/SIGMA/kuznetsov.html
2 N. Iorgov, V. Roubtsov, V. Shadura and Yu. Tykhyy
The goal of this contribution is to find common eigenvectors of commuting Hamiltonians of
the RTC at a root of unity with boundary interaction. We use for this the approach from [5].
Let us describebriefly the structureof thepaper. We start with ashortaccount of the model
under consideration and remind its properties based on the presentation in [11] (Section 2). We
recallthestandardSklyaninformalismintroducingachainofL-operatorssatisfyingsomeRLL =
LLR relations as well as its “twisted” version (2.3) such that the corresponding (“twisted”)
monodromy matrix satisfies the “Reflection Equations” and some “unitarity” condition. We
specifya“boundarymatrix”solutionK (λ)ofthedualReflection Equations whicharenecessary
+
for a description of a two-boundary RTC. We close this section with a Hamiltonian generating
function to the two-boundary RTC.
We give the explicit formulas for the eigenvectors of the open RTC in the Section 3 using the
eigenvectors formulas for the open Bazhanov–Stroganov quantum chain of which the open RTC
is a (specified) particular case [9].
In Section 4 we consider the eigenvectors of the one-boundary RTC specifying some par-
ticular choice of the boundary matrices. We identify the coefficients of the corresponding
generating function with the commuting one-boundary RTC Hamiltonians and compute the
common eigenvectors. The same questions are treated in Section 5 for the two-boundary RTC
using the Separation of Variables method. We write the Baxter equation for separated variables
explicitly (5.3).
Section 6 contains an another proof of the commutativity of the one-boundaryRTC Hamilto-
nians of the Section 4. We show in fact that they belong to a wide class of Hamiltonians which
can be expressed like a sort of Gelfand–Retakh quasideterminants. This quasideterminants are
(in our case) just Plu¨cker coordinates ∆ ∆−1 on Gr (n). The commutation (and, hence the
s 0 n−1
existence of the common spectrum) is guaranteed by the results of [12] (see also [13, 14]).
We finish with a short discussion in the Section 7 and we leave some tedious computations
to the Appendix A.
WearededicatingthispapertothememoryofVadimKuznetsov. Hisimportantcontributions
to the subject of Separation of Variables and in particularly to its applications in quantum
integrable models with boundary interaction [4, 15, 16, 17] are widely acknowledged in the
Mathematical Physics community. In particular, it was Kuznetsov’s paper [16] where the Se-
paration of Variables was performed for the quantum relativistic Toda chain and commuting
Hamiltonians of the quantum RTC with integrable boundary interaction (the quantum RTC for
general classical root systems) were obtained.
2 The formulation of the model
In this section we recall briefly the subject of the model called the quantum Relativistic Toda
Chain (RTC) chain at root of unity [11].
Let ω = e2πi/N, N 2. For each particle k, k = 1,2,...,n, of the n-particle RTC there is
≥
a corresponding N-dimensional linear space (quantum space) with the basis γ ,γ Z ,
k k N
V | i ∈
and a pair of operators u ,v acting on by the formulas:
k k k
{ } V
v γ = vωγ γ , u γ = uγ 1 .
k k k k k k
| i | i | i | − i
The space of quantum states of RTC with n particles is = . We extend
1 2 n
V V ⊗V ⊗···⊗V
the action of operators u ,v to defining this action to be identical on , s = k. Thus we
k k s
{ } V V 6
have the following commutation relations
u u = u u , v v = v v , u v = ωδj,kv u .
j k k j j k k j j k k j
Relativistic Toda Chain with Boundary Interaction at Root of Unity 3
Monodromy matrix for RTC with n particles is
A (λ) B (λ)
T (λ) = L (λ)L (λ) L (λ) = n n ,
n 1 2 ··· n Cn(λ) Dn(λ)
(cid:18) (cid:19)
where
1+ κv 1u
L (λ) = λ k −λ k , k = 1,2,...,n. (2.1)
k u−k1vk 0 !
Thus the modeldependson three parameters u, v and κ. It is possibleto make this model to be
inhomogeneous by attaching the index k to the parameters u, v and κ. All the results presented
in this paper can be directly generalized to inhomogeneous case.
The straightforward calculation shows that the operators L (λ) satisfy the following commu-
k
tation relations:
(1) (2) (2) (1)
R(λ,µ)L (λ)L (µ) = L (µ)L (λ)R(λ,µ), (2.2)
k k k k
where the standard notations L(1)(λ) = L (λ) I, L(2)(µ) = I L (µ) were used and the
k k ⊗ k ⊗ k
R-matrix R(λ,µ) has the form
λ ωµ 0 0 0
−
0 λ µ (1 ω)µ 0
R(λ,µ) = − − .
0 (1 ω)λ ω(λ µ) 0
− −
0 0 0 λ ωµ
−
Using (2.2), we obtain the following relation for monodromy matrix
R(λ,µ)T(1)(λ)T(2)(µ) = T(2)(µ)T(1)(λ)R(λ,µ).
n n n n
We will use the method [6] of Sklyanin for description of RTC with integrable boundary
interaction. The L-operator (2.1) together with
κλ+ωθv−1 θu−1
L (λ) = λσ L−1(θ/λ)σ = k − k = ωθv−1σ L (ωθ/λ)Tσ (2.3)
k x k x λvk−1uk 0 ! k z k z
e
satisfy the following relations
(1) (2) (2) (1)
R(λ,µ)L (λ)L (µ) = L (µ)L (λ)R(λ,µ),
k k k k
(1) (2) (2) (1)
R(λ,µ)L (λ)L (µ) = L (µ)L (λ)R(λ,µ),
k k k k
(2) (1) (1) (2)
L (µ)R(λ,µ)L (λ) = L (λ)R(λ,µ)L (µ),
k e ek ek e k
(2) (1) (1) (2)
L (µ)R(λ,µ)L (λ) = L (λ)R(λ,µ)L (µ),
ek e k k e ek
where e e e e
θ λµ 0 0 ωθ θ
− −
0 ωθ λµ 0 0
R(λ,µ) = µσ(2)R(λ,θ/µ)σ(2) = − ,
x x 0 0 ωθ λµ 0
−
ωλµ λµ 0 0 ωθ ωλµ
e − −
0 1 1 0
σ = , σ = .
x 1 0 z 0 1
(cid:18) (cid:19) (cid:18) − (cid:19)
4 N. Iorgov, V. Roubtsov, V. Shadura and Yu. Tykhyy
Hereθ isanarbitraryparameter. ItisimportantthatR(µ,λ) = R(λ,µ) = P R(λ,µ)P ,where
12 12
the operator P is the operator of permutation of two spaces. Then the matrix
12
e e e
(λ) (λ)
U(λ) = Ln(λ)···L1(λ)K−(λ)L1(λ)···Ln(λ) = A(λ) B(λ)
(cid:18) C D (cid:19)
e e
satisfies the reflection equation
R(λ,µ)U(1)(λ)R(λ,µ)U(2)(µ) = U(2)(µ)R(λ,µ)U(1)(λ)R(λ,µ) (2.4)
if K (λ) satisfies theesame equation: e
−
(1) (2) (2) (1)
R(λ,µ)K (λ)R(λ,µ)K (µ) = K (µ)R(λ,µ)K (λ)R(λ,µ). (2.5)
− − − −
The solution K (λ) weith non-operator matrixeelements is
−
α (θ λ) η +β θ
K−(λ) = η−+λβ−λ δ−(θ −λλ) .
(cid:18) − − − λ − (cid:19)
This solution depends on four parameters α , β , δ , η . It can be found directly solving (2.5)
− − − −
or adapting the formulas from [18] (see also [16]). The matrix U(λ) satisfies the so-called
unitarity condition:
(ωθ/λ) (ωθ/λ)
U(ωθ/λ) = A B (2.6)
(ωθ/λ) (ωθ/λ)
(cid:18) C D (cid:19)
1 (λ2 ω2θ) (λ) λ2(1 ω) (λ)+(ωθ λ2) (λ)
= − A − B − C .
ω(θ λ2) ω(ωθ λ2) (λ)+ωθ(1 ω) (λ) (λ2 ω2θ) (λ)
− (cid:18) − B − C − D (cid:19)
We have (λ) (µ) = (µ) (λ) from the reflection equation (2.4). It means that (λ)
A A A A A
is generation function with respect to λ for commuting set of operators. These operators are
Hamiltonians for one-boundary RTC.
To describe a two-boundary RTC we need the dual reflection equation:
(2) (1) (1) (2)
R(λ,µ)K (µ)Q(λ,µ)K (λ) = K (λ)Q(λ,µ)K (µ)R(λ,µ),
+ + + +
where
Q(λ,µ) = Rt1(λ,µ) −1 t1.
(cid:0)(cid:0) (cid:1) (cid:1)
Its solution is e
α (ω2θ λ) η +β ωθ
K+(λ) = η+ω+λ β−λ δ+(ωθ + ωλ−1λ) .
(cid:18) + + + λ − (cid:19)
Then the generation function for the Hamiltonians of the two-boundary RTC is given by t(λ) =
Tr(K (λ)U(λ)) or explicitly:
+
t(λ) = α (θω2/λ λ) (λ)+(β λ+η ω) (λ)
+ + +
− A B
+(β θω/λ+η ) (λ)+δ (ωθ/λ ω−1λ) (λ).
+ + +
C − D
Relativistic Toda Chain with Boundary Interaction at Root of Unity 5
3 Eigenvectors for the open RTC
In order to give explicit formulas for the eigenvectors for RTC we remind the definition (see
for example [19]) of w (s) which is an analogue of Γ -function at root of unity. For any point
p q
p = (x,y) of Fermat curve xN +yN = 1, we define w (s), s Z , by
p N
∈
w (s) y
p
= , w (0) = 1. (3.1)
w (s 1) 1 xωs p
p
− −
Due to the Fermat curve condition the function w (s) is cyclic: w (s+N) = w (s).
p p p
The explicit formulas for the eigenvectors of open RTC, that is eigenvectors of A (λ), can be
n
extracted from [8], where the formulas for the eigenvectors of open Bazhanov–Stroganov quan-
tum chain (which includes RTC) were obtained. Here we give the resulting formulas specialized
to open RTC with a little change of notations.
The eigenvectors of open RTC are given in terms of w (s) defined by the different points
p
p = (x ,y ),m,m′ = 1,2,...,n;s = 1,2,...,m;s′ = 1,2,...,m′,belonging
m,s;m′,s′ m,s;m′,s′ m,s;m′,s′
to Fermat curve xN + yN = 1. The x-coordinates of these points are defined by unknown
parameters (amplitudes) v by x = v /v . Then the corresponding y are
m,s m,s;m′,s′ m,s m′,s′ m,s;m′,s′
defined up to a root of 1. The parameters v are defined by the initial value v := v and the
m,s 11
recurrent relations
v v v = v v v v,
m1 m2 mm m−1,1 m−1,2 m−1,m−1
··· ···
m
y
m−1,l;m,k
y
κ m−1,s;m−1,l k=1 =1, l = 1,2,...,m 1.
y m−Q2 −
m−1,l;m−1,s
s6=l y
Y m−2,s;m−1,l
s=1
Q
In the case of homogeneous RTC these relations can be solved explicitly [11] (see also [8]).
We will use the notation γ for the eigenvectors of the operator A (λ) of
n 1 n n
| i ∈ V ⊗···⊗V
the open RTC with n particles. These eigenvectors are labelled by n parameters γ Z ,
n,s N
∈
s = 1,2,...,n, collected into a vector
γ = (γ ,...,γ ) (Z )n.
n n,1 n,n N
∈
The formula
n n−1
γ = µ(γ )Q(γ γ )γ γ γ ,
n n−1 n−1 n n−1 n,k n−1,l n
| i | | i⊗| − i
γXn−1 Xk=1 Xl=1
where
n−1
µ(γ ) = w (γ γ 1),
n−1 pn−1,j;n−1,l n−1,j − n−1,l−
j,l=1
Y
(j6=l)
ωγn−1,1+···+γn−1,n−1
Q(γ γ ) = ,
n−1 n
| n n−1
w (γ γ 1)
pn−1,l;n,k n−1,l− n,k −
k=1l=1
Q Q
gives the eigenvectors γ of A (λ) from the eigenvectors γ of A (λ)
n n n−1 1 n−1 n−1
| i | i ∈ V ⊗···⊗V
and basis vectors γ . To find the formula for γ we can use the same formula and
n n n−1
| i ∈ V | i
6 N. Iorgov, V. Roubtsov, V. Shadura and Yu. Tykhyy
so on. At the last step we need the eigenvectors of 1-particle RTC which are just basis vectors
γ . The vectors γ obtained by the described procedure satisfy
1,1 1 1 n
| i ∈ V | i
n κv ωγn,k n λ
n,k n,k
A (λ)γ = 1+ γ = 1 γ ,
n n n n
| i λ | i − λ | i
k=1(cid:18) (cid:19) k=1(cid:18) (cid:19)
Y Y
h
B (λ )γ = k γ−k ,
n n,k | ni −λ | n i
n,k
n
h λ (λ λ )
B (λ)γ = k n,k − n,s γ−k ,
n | ni − λ λ(λ λ ) | n i
k=1s6=k(cid:18) n,k − n,s (cid:19)
XY
n−1
where λn,k = −κvn,kωγn,k and hk = u yn−1,l;n,k. In the above formulas the vector |γn−ki
l=1
means the vector γn1,...,γnn with γn,kQreplaced by γn,k 1.
| i −
We will omit in what follows the index n in matrix elements of monodromy matrix.
At λ = ωµ, the rank of R(λ,µ) becomes 1. Thereforeboth sides of (2.2) become proportional
to R(λ,µ) with the coefficient being the so-called quantum determinant:
V
qdet T (µ) = D(ωµ)A(µ) B(ωµ)C(µ)= qdetL (µ) qdet L (µ)= , (3.2)
n − 1 ··· n (ωµ)n
where
n n
V = v , V γ = (v ωγn,k) γ .
k n n,k n
| i | i
k=1 k=1
Y Y
Now use (3.2) for µ = λ :
n,r
V
D(ωλ )A(λ ) B(ωλ )C(λ ) = .
n,r n,r − n,r n,r (ωλ )n
n,r
Acting by both sides on γ we get
n
| i
n
(v ωγn,k)
n,k
B(ωλ )C(λ )γ = k=1 γ .
− n,r n,r | ni Q(ωλ )n | ni
n,r
Therefore
n
(v ωγn,k)
n,k
C(λ )γ = k=1 γ+r .
n,r | ni (Qωλ )n−1h | n i
n,r r
Since C(λ) is a polynomial in 1/λ of degree n 1 we can reconstruct the action of C(λ):
−
n
λ (λ λ )
n,r n,m
C(λ)γ = − C(λ )γ .
n n,r n
| i λ(λ λ ) | i
n,r n,m
r=1 m6=r −
X Y
Finally using (3.2) after some calculation we get
ωκ n vn,mωγn,m v ωγn,k
D(λ)γ = m n,k
| ni λ r=1k6=r"Q(−κω)n (cid:18)(vn,kωγn,k −ωvn,rωγn,r)(vn,rωγn,r −vn,kωγn,k)(cid:19)
XX
vn,kωγn,k(λ+κvn,sωγn,s) hk γ+r−k γ .
×s6=k,r(cid:18)λ(vn,kωγn,k −vn,sωγn,s)(vn,rωγn,r −vn,sωγn,s)(cid:19)(cid:18)hr| n i−| ni(cid:19)#
Y
Relativistic Toda Chain with Boundary Interaction at Root of Unity 7
4 Eigenvectors for the one-boundary RTC
We restrict our attention in what follows to the choice β = δ = 0. In this case the boundary
± ±
matrices become
α (θ λ) η α (ω2θ λ) η
K−(λ) = − λ − − , K+(λ) = + λ − + .
η 0 η ω 0
(cid:18) − (cid:19) (cid:18) + (cid:19)
Due to (2.3) we have
A(ωθ/λ) C(ωθ/λ)
Ln(λ)···L1(λ) = (ωθ)nV−1σzTt(ωθ/λ)σz = (ωθ)nV−1 B(ωθ/λ) D−(ωθ/λ) .
(cid:18) − (cid:19)
e e
Therefore
θ λ2
(λ) = (ωθ)nV−1 α − A(ωθ/λ)A(λ)+η (A(ωθ/λ)C(λ) C(ωθ/λ)A(λ)) ,
− −
A λ −
(cid:20) (cid:21)
θ λ2
(λ) = (ωθ)nV−1 α − A(ωθ/λ)B(λ)+η (A(ωθ/λ)D(λ) C(ωθ/λ)B(λ)) ,
− −
B λ −
(cid:20) (cid:21)
θ λ2
(λ) = (ωθ)nV−1 α − B(ωθ/λ)A(λ)+η ( B(ωθ/λ)C(λ)+D(ωθ/λ)A(λ)) , (4.1)
− −
C − λ −
(cid:20) (cid:21)
θ λ2
(λ) = (ωθ)nV−1 α − B(ωθ/λ)B(λ)+η ( B(ωθ/λ)D(λ)+D(ωθ/λ)B(λ)) .
− −
D − λ −
(cid:20) (cid:21)
From (2.6) it follows that (λ) defined by (λ) = λ (λ)/(θ λ2), or explicitly
A A A −
η λ
(λ) = (ωθ)nV−1 αeA(ωθ/λ)A(λ)+e − (A(ωθ/λ)C(λ) C(ωθ/λ)A(λ)) , (4.2)
A − (θ λ2) −
(cid:20) − (cid:21)
e
satisfies (ωθ)= (λ). It means that (λ) can be presented as
A λ A A
A(λe) = α−κne (λ+ωθ/λ)n+(λ+eωθ/λ)n−1H1B+···+(λ+ωθ/λ)HnB−1+HnB , (4.3)
whereethe set of op(cid:0)erators H1B,H2B,...,HnB have to be identified with the commut(cid:1)ing set of
Hamiltonians for the one-boundary RTC. Explicitly, HB is given by (5.2) with η = 0. Our
1 +
problem is to find the common eigenvectors for this set.
For σ = (σ1,...,σn) (ZN)n, let
∈
Ψ = Q(γ ,σ)γ , (4.4)
σ n n
| i
Xγn
where the sum includes all Nn combinations of γ = γ ,...,γ and
n n,1 n,n
{ }
n n n
Q(γn,σ) = ωγn2,r+γn,r wpBr,k(γn,r −σk)wp˜Br,k(γn,r +σk)
rY=1 kY=1rY=1(cid:16) (cid:17)
vn,rωγn,r vn,r′ωγn,r′
− . (4.5)
×r<r′ wp˜r,r′(γn,r +γn,r′) !
Y
Then, as it will be shown in the Appendix A, we have
n (λ+κs ωσk)(ωθ/λ+κs ωσk)
k k
(λ)Ψ = α Ψ
A σ −k=1(cid:18) skωσk (cid:19) σ
Y
e
8 N. Iorgov, V. Roubtsov, V. Shadura and Yu. Tykhyy
n
(λ λ )(ωθ/λ λ )
= α ( κ)n − k − k Ψ , (4.6)
− σ
− λ
k=1(cid:18) k (cid:19)
Y
where s , k = 1,2,...,n, are some fixed amplitudes which will be defined later. Also we have
k
used a short notation λ = κs ωσk. This form of the eigenvalue polynomial corresponds
k k
−
to (4.3). We are using three types of points p˜ , pB and p˜B belonging to the Fermat curve
r,r′ r,k r,k
xN +yN = 1 in the definition of Q(γ ,σ). They are defined by their x-coordinates:
n
x˜ = κ2v v /(ωθ), xB = v /s , x˜B = κ2v s /(ωθ)
r,r′ n,r n,r′ r,k n,r k r,k n,r k
and the equations
η v2 n
− n,r y˜ = α h yB y˜B , r = 1,2,...,n. (4.7)
κn−2ωn+1θ r,r′ − r r,k r,k
r′6=r k=1
Y Y
To solve these equations we have to take the N-th degree of both parts of the equations (4.7)
and solve them with respect to elementary symmetric polynomials of sN, k = 1,2,...,n. This
k
information is enough to find the values of s and the coordinates of the Fermat points entering
k
Q(γ ,σ).
n
Taking into account Laurent polynomial structure of (λ) in λ it is clear that it is enough
C
to know the action formulas for (λ) on Ψ in 2n+2 points of λ to reconstruct (λ)Ψ . The
σ σ
C C
following formulas are valid:
(λ )Ψ = η f Ψ , (ωθ/λ )Ψ = η (ωθ)nf−1Ψ , (4.8)
C k σ − k σ+k C k σ − k σ−k
( √θ)Ψ = η ( √θ)nΨ , (4.9)
σ − σ
C ± ±
n
where λ = κs ωσk, f = λ yB /y˜B . The reconstructed formula for (λ) is
k − k k k r,k r,k C
r=1
Q
θ λ2 n λ (λ λ )(ωθ λλ )
k r r
(λ)Ψ = − − −
σ
C λ λ(λ λ )(ωθ λ λ )
k r k r
k=1r6=k − −
XY
λ ωθ λλ ωλ (λ λ )
k k k k
− (λ )Ψ − (ωθ/λ )Ψ
× ωθ λ2 θ λ2 C k σ− ω2θ λ2 C k σ
− k (cid:18) − k − k (cid:19)
n
1 (λ λ )(ωθ λλ )
+ − r − r (1+ǫλ/√θ) (ǫ√θ)Ψ . (4.10)
σ
2 λ(ǫ√θ λ )(ωǫ√θ λ ) C
ǫX=±1rY=1 − r − r
Here we give some heuristic explanation of formulas (4.8). Let Ψ be an eigenvector function
σ
of (λ) satisfying (4.6). Then acting by both sides of commutation relation
A
(ωµ µ)(θ λµ) (λ) (µ)+(µ λ)(ωθ λµ) (λ) (µ)
− − C A − − A C
+(µ λ)(ωλµ λµ) (λ) (µ) = (θ λµ)(ωµ λ) (µ) (λ),
− − B A − − C A
which follows from (2.4), on Ψ at µ = λ we get
σ k
(λ ωλ )(θ/λ λ )
k k
(λ) (λ )Ψ = − − (λ ) (λ)Ψ .
k σ k σ
A C (λ λ )(θω/λ λ )C A
k k
− −
Using (4.6) we see that (λ )Ψ is an eigenvector of (λ) with shifted zero, that is λ replaced
k σ k
C A
by ωλ . It means that (λ )Ψ is proportional to Ψ . Clearly, this argumentation is not
k k σ σ+k
C
sufficient to prove the relations (4.8). The derivation of these relations can be carried out
adapting the corresponding derivation from [5]. To prove (4.9) we use (4.1) at λ = √θ and
±
then (3.2).
Relativistic Toda Chain with Boundary Interaction at Root of Unity 9
5 Eigenvectors for the two-boundary RTC.
Separation of variables
From (2.6) it follows
(λ)(ω 1)θ+ (ωθ)(θ λ2)
(λ) = C − C λ − ,
B ωθ λ2
−
which gives
ω2θ λ2
t(λ) = − α (λ)+η (λ)+η ω (λ)
+ + +
λ A C B
(ω2θ λ2)(θ λ2) (λ)(ω2θ λ2)+ω (ωθ)(θ λ2)
= − − α (λ)+η C − C λ − . (5.1)
λ2 +A + ωθ λ2
−
We read off the structure of t(λ) froem this expression
α α κn
t(λ) = (ω2θ2/λ2 θ)(λ2 θ) + − (λ+ωθ/λ)n+(λ+ωθ/λ)n−1HBB +
− − − θ 1 ···
+(λ+ωθ/λ)HBB +HBB +η η(cid:0) p (λ),
n−1 n + − n
(cid:1)
where
p (λ) = (λ+ωθ/λ)θ(n−1)/2 if n is odd, p (λ) = (1+ω)θn/2 if n is even
n n
and the set of operators HBB,HBB,...,HBB should be identified with the commuting set of
1 2 n
Hamiltonians for the two-boundary RTC. Explicitly,
n n−1
HBB = κ v +ωθκ−2v−1 u u−1 v +ωθκ−2u u−1 v−1
1 k k − k k+1 k+1 k k+1 k
k=1 k=1
X(cid:0) (cid:1) X(cid:0) (cid:1)
(η /α )u−1v (η /α )κ−1v−1u . (5.2)
− − − 1 1− + + n n
Let ΦE, E = (E1,...,En), be a common eigenvector for the commuting set of operators
HBB :
{ k }
HkBBΦE = EkΦE
and tE(λ) is corresponding eigenvalue polynomial for t(λ). In accordance with a procedure of
the Separation of Variables we are looking for ΦE to be in the form
n
ΦE = (λi λj)(ωθ λiλj) qk(σk,E)Ψσ,
− −
σ i<j k=1
XY(cid:0) (cid:1)Y
where λ = κs ωσk.
k k
−
Using the action formula (5.1) for t(λ) on Ψ together with (4.6) and (4.10) we get Baxter
σ
equations for separated variables q (σ ,E):
k k
(θ λ2)
tE(λk)qk(σk,E) = η+η−(ωθ −ω−k2λ2)ω1−2nfkqk(σk −1,E)
− k
(ω2θ λ2)
+η η − k ωn−1θnf−1q (σ +1,E), (5.3)
+ − (θ ωλ2) k k k
− k
10 N. Iorgov, V. Roubtsov, V. Shadura and Yu. Tykhyy
n
where f = λ yB /y˜B . The corresponding Baxter equation can be considered (for each
k k r,k r,k
r=1
k = 1,2,...,nQ) as a homogeneous system of linear equations for N unknowns qk(σ,E), σ ZN.
∈
The requirement of the existence of non-trivial solutions for all k = 1,2,...,n fixes the common
spectrum for the commuting set of operators HBB . Unfortunately, this problem can not be
{ k }
solved explicitly for general N. But if we know the spectrum we can construct the eigenvectors
solving the systems of linear equations. We would like to report about such a progress [7] in the
case of the periodic Baxter–Bazhanov–Stroganov model at N = 2 which contains in particular
the Ising model.
B
6 Another proof of commutativity of H
k
InthissectionwedescribeanalternativeapproachtotheproofofcommutativityofHamiltonians
usingas an examplethesetof theone-boundaryRTCHamiltonians HB. Thisapproach is based
k
on the results from [12]. Let us consider the generation function
n n
(λ+v˜ )(θ/λ+v˜ ) (λ+v˜ )(θ/λ+v˜ )
(λ) = α k k +η u˜−1v˜ s s
A v˜ r r v˜ v˜
k r s
k=1 k=1 s6=k −
Y X Y
n n
(Λ+v˜ +θ/v˜ )
= α (Λ+v˜ +θ/v˜ )+η u˜−1V˜ s s , (6.1)
k k k v˜ v˜
k s
k=1 k=1 s6=k −
Y X Y
where Λ= λ+θ/λ, V˜ = v˜ v˜ v˜ . The set of operators u˜ , v˜ satisfy the relations
1 2 n j j
···
u˜ u˜ = u˜ u˜ , v˜ v˜ = v˜ v˜ , u˜ v˜ = qδj,kv˜ u˜ ,
j k k j j k k j j k k j
where q is a non-zero complex number. Note that this expression for (λ) coincides with (4.2)
A
(see also (A.3)) up to a redefinition of parameters.
Expanding in Λ we obtain
A
= α(Λn+HBΛn−1+ +HB Λ+HB). (6.2)
A 1 ··· n−1 n
We will prove that the operators HB obtained by such expansion are pairwise commuting. Let
k
us introduce new operators Λ = v˜ θ/v˜ . Taking into account
k k k
− −
Λ Λ = (v˜ v˜ )(θ/(v˜ v˜ ) 1),
k s k s k s
− − −
we get
n n
Λ Λ
= α (Λ Λ )+η Z˜ − s ,
k k
A − Λ Λ
k s
k=1 k=1 s6=k −
Y X Y
where
v˜2
Z˜ = u˜−1v˜n s .
k k k (θ v˜ v˜ )
k s
s6=k −
Y
n
Finally using the interpolation formula for the polynomial (Λ Λ ) Λn of degree n 1:
k
− − −
k=1
Q
n n
Λ Λ
(Λ Λ ) = Λn Λn − s ,
− k − k Λ Λ
k s
k=1 k=1 s6=k −
Y X Y
