Eigenvectors of Open Bazhanov-Stroganov Quantum Chain
In this contribution we give an explicit formula for the eigenvectors of Hamiltonians of open Bazhanov-Stroganov quantum chain. The Hamiltonians of this quantum chain is defined by the generation polynomial An(λ) which is upper-left matrix element of monodromy matrix built from the cyclic L-operator...
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irk-123456789-1464242019-02-10T01:23:35Z Eigenvectors of Open Bazhanov-Stroganov Quantum Chain Iorgov, N. In this contribution we give an explicit formula for the eigenvectors of Hamiltonians of open Bazhanov-Stroganov quantum chain. The Hamiltonians of this quantum chain is defined by the generation polynomial An(λ) which is upper-left matrix element of monodromy matrix built from the cyclic L-operators. The formulas for the eigenvectors are derived using iterative procedure by Kharchev and Lebedev and given in terms of wp(s)-function which is a root of unity analogue of Γq-function. 2006 Article Eigenvectors of Open Bazhanov-Stroganov Quantum Chain / N. Iorgov // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 17 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81R12; 81R50 http://dspace.nbuv.gov.ua/handle/123456789/146424 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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In this contribution we give an explicit formula for the eigenvectors of Hamiltonians of open Bazhanov-Stroganov quantum chain. The Hamiltonians of this quantum chain is defined by the generation polynomial An(λ) which is upper-left matrix element of monodromy matrix built from the cyclic L-operators. The formulas for the eigenvectors are derived using iterative procedure by Kharchev and Lebedev and given in terms of wp(s)-function which is a root of unity analogue of Γq-function. |
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Iorgov, N. |
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Iorgov, N. Eigenvectors of Open Bazhanov-Stroganov Quantum Chain Symmetry, Integrability and Geometry: Methods and Applications |
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Iorgov, N. |
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Iorgov, N. |
| title |
Eigenvectors of Open Bazhanov-Stroganov Quantum Chain |
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Eigenvectors of Open Bazhanov-Stroganov Quantum Chain |
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Eigenvectors of Open Bazhanov-Stroganov Quantum Chain |
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Eigenvectors of Open Bazhanov-Stroganov Quantum Chain |
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Eigenvectors of Open Bazhanov-Stroganov Quantum Chain |
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eigenvectors of open bazhanov-stroganov quantum chain |
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Інститут математики НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/146424 |
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Eigenvectors of Open Bazhanov-Stroganov Quantum Chain / N. Iorgov // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 17 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT iorgovn eigenvectorsofopenbazhanovstroganovquantumchain |
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2025-07-10T23:58:03Z |
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Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 019, 10 pages
Eigenvectors of Open Bazhanov–Stroganov
Quantum Chain
Nikolai IORGOV
Bogolyubov Institute for Theoretical Physics, 14b Metrolohichna Str., Kyiv, 03143 Ukraine
E-mail: iorgov@bitp.kiev.ua
Received November 29, 2005, in final form January 30, 2006; Published online February 04, 2006
Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper019/
Abstract. In this contribution we give an explicit formula for the eigenvectors of Hamil-
tonians of open Bazhanov–Stroganov quantum chain. The Hamiltonians of this quantum
chain is defined by the generation polynomial An(λ) which is upper-left matrix element
of monodromy matrix built from the cyclic L-operators. The formulas for the eigenvec-
tors are derived using iterative procedure by Kharchev and Lebedev and given in terms of
wp(s)-function which is a root of unity analogue of Γq-function.
Key words: quantum integrable systems; Bazhanov–Stroganov quantum chain
2000 Mathematics Subject Classification: 81R12; 81R50
1 Introduction
In the papers [1, 2] it was observed that the six-vertex R-matrix at root of unity intertwines
not only the six-vertex L-operators, but also some other L-operators (which are called cyclic
L-operators). These L-operators define n-particle Bazhanov–Stroganov quantum chain (BSQC)
by the standard procedure in quantum inverse scattering method: the product of L-operators
defines the monodromy matrix
Tn(λ) = L1(λ)L2(λ) · · ·Ln(λ) =
(
An(λ) Bn(λ)
Cn(λ) Dn(λ)
)
, (1)
which in turn provides us with commuting set of operators — Hamiltonians of quantum chain.
It was observed by Baxter that the so-called “Inverse SOS” model discovered by him [3] is
equivalent to Bazhanov–Stroganov quantum chain. Thus the same model (which is called the
τ2-model) has two formulations: the formulation as face model by Baxter and the formulation
as quantum chain (or vertex model) by Bazhanov and Stroganov. The connection between six-
vertex model, τ2-model and chiral Potts model gave a possibility to formulate the system of
functional relations [1, 4] for transfer-matrices of these models. This system provides the main
tool for derivation the free energy [5] and the order parameter [6] for chiral Potts model.
The goal of this contribution is to find common eigenvectors of the set of commuting Hamil-
tomians Hk, k = 1, 2, . . . , n, of open n-particle BSQC. These Hamiltonians are defined by the
coefficients of the An(λ) given by (1):
An(λ) = 1 + λH1 + λ2H2 + · · ·+ λnHn.
The main idea how to find the eigenvectors is to use iterative procedure. Namely we build
the eigenvectors of An(λ) using the eigenvectors of An−1(λ) which is the generation function
of Hamiltonians for open (n − 1)-particle BSQC. This procedure in essential is an adaptation
mailto:iorgov@bitp.kiev.ua
http://www.emis.de/journals/SIGMA/2006/Paper019/
2 N. Iorgov
of iterative procedure by Kharchev and Lebedev [7] for quantum Toda chain. The mentioned
idea has origin in the paper by Sklyanin [8], where he used separated variables of subsystems to
construct the separated variables of the whole system. The models where the iterative procedure
was realized are relativistic Toda chain [9], Toda chain with boundary interaction [10], periodic
Bazhanov–Stroganov model [11]. It is worth to note that these models do not admit algebraic
Bethe ansatz procedure because in the case of generic parameters these models do not possess
“highest weight vectors”. The method used in this contribution is an evolution of the method
of separated variables (or functional Bethe ansatz method).
At the end of the introduction we would like to mention that the Bazhanov–Stroganov model
at special values of parameters reduces to the relativistic Toda chain at a root of unity. Also
it is worth noting direct (not through the chiral Potts model!) relation [12] between lattice
formulation of the model at N = 2 and Ising model. This relation gave a possibility to find the
eigenvalues [12] and the eigenvectors [13] of the transfer-matrix by means of auxiliary grassmann
field technique.
2 Bazhanov–Stroganov quantum chain
Let ω = e2πi/N , N ≥ 2. For each particle k, k = 1, 2, . . . , n, of Bazhanov–Stroganov quantum
chain (BSQC) with n particles there corresponds N -dimensional linear space (quantum space) Vk
with the basis |γ〉k,γ ∈ ZN , and a pair of operators {uk,vk} acting on Vk by the formulas:
vk|γ〉k = ωγ |γ〉k, uk|γ〉k = |γ − 1〉k. (2)
The space of quantum states of BSQC with n particles is V = V1 ⊗ V2 ⊗ · · · ⊗ Vn. We extend
the action of operators {uk,vk} to V defining this action to be identical on Vs, s 6= k. Thus we
have the following commutation relations
ujuk = ukuj , vjvk = vkvj , ujvk = ωδj,kvkuj .
For each particle of BSQC model we put into correspondence the cyclic L-operator
Lk(λ) =
(
1 + λκkvk λu−1
k (ak − bkvk)
uk(ck − dkvk) λakck + vkbkdk/κk
)
, k = 1, 2, . . . , n, (3)
where {ak, bk, ck, dk, κk} are (in general complex) parameters attached to kth particle. In the
papers [1, 2] it was observed that the six-vertex R-matrix
R(λ, µ) =
λ− ωµ 0 0 0
0 ω(λ− µ) λ(1− ω) 0
0 µ(1− ω) λ− µ 0
0 0 0 λ− ωµ
.
at root of unity ω = e2πi/N intertwines not only the six-vertex L-operator, but also the cyclic
L-operators (3):
R(λ, µ) L
(1)
k (λ)L(2)
k (µ) = L
(2)
k (µ) L
(1)
k (λ) R(λ, µ),
where L
(1)
k (λ) = Lk(λ) ⊗ I, L
(2)
k (µ) = I ⊗ Lk(µ). In fact the formulas for L-operators and R-
matrix given in this contribution are close to formulas from the paper by Tarasov [14]. Original
formulas given in [1] and [2] are a bit different (but equivalent). The monodromy matrix for the
BSQC with n particles is defined as
Tn(λ) = L1(λ)L2(λ) · · ·Ln(λ) =
(
An(λ) Bn(λ)
Cn(λ) Dn(λ)
)
(4)
Eigenvectors of Open Bazhanov–Stroganov Quantum Chain 3
and satisfies the same intertwining relation
R(λ, µ) T (1)
n (λ) T (2)
n (µ) = T (2)
n (µ) T (1)
n (λ) R(λ, µ). (5)
The intertwining relation (5) gives [An(λ), An(ν)] = 0. Therefore An(λ) is the generating
function for the commuting set of operators H1, . . . ,Hn:
An(λ) = 1 + λH1 + λ2H2 + · · ·+ λnHn.
We interpret these operators H1, . . . ,Hn as Hamiltonians of the open BSQC. The simplest
Hamiltonians are
H1 =
n∑
k=1
κkvk +
∑
1≤l<k≤n
u−1
l (al − blvl)
k−1∏
s=l+1
bsds
κs
vs · uk(ck − dkvk), Hn =
n∏
k=1
κkvk.
At bk = 0 and ck = 0, the BSQC model reduces to Relativistic Toda Chain (RTC) at root of
unity. The corresponding L-operators are
LRTC
k (λ) =
(
1 + λκkvk λaku
−1
k
−dkukvk 0
)
, k = 1, 2, . . . , n.
As in BSQC model ARTC
n (λ) is the generating function for the commuting set of operators
HRTC
1 , . . . ,HRTC
n . The simplest Hamiltonians for RTC are
HRTC
1 =
n∑
k=1
κkvk −
∑
1≤l≤n−1
aldl+1u
−1
l ul+1vl+1, HRTC
n =
n∏
k=1
κkvk.
Note that these operators HRTC
1 , . . . ,HRTC
n are the Hamiltonians of the RTC with open boun-
dary condition and association with the standard operators of momenta pk and positions qk
roughly speaking (up to constants) is vk = exppk and uk = exp qk. Then HRTC
n is the exponent
of the total momentum of RTC and HRTC
1 is the Hamiltonian of relativistic analogue of usual
Toda chain.
3 Eigenvalues and associated amplitudes
In this section we give a procedure of obtaining the eigenvalues for open n-particle BSQC
Hamiltonians Hk, k = 1, 2, . . . , n, or equivalently for An(λ).
In the case of ak = bk = ck = dk = 0, k = 1, 2, . . . , n, we have An(λ) =
n∏
k=1
(1 + κkvk).
We interpret the corresponding Hamiltonians as free (without interaction between particles)
Hamiltonians. Due to (2) the standard basis vectors |γ1, γ2, . . . , γn〉 = |γ1〉1⊗|γ2〉2⊗· · ·⊗|γn〉n ∈
V are eigenvectors of An(λ):
An(λ)|γ1, γ2, . . . , γn〉 =
n∏
k=1
(1 + κkω
γk)|γ1, γ2, . . . , γn〉.
We claim that in the general case the spectrum of An(λ) has the form as in the case of
non-interacting particles but with modified amplitudes κn,k:
An(λ)|γ1, γ2, . . . , γn〉 =
n∏
k=1
(1 + κn,kω
γkλ) |γ1, γ2, . . . , γn〉, γk ∈ ZN .
4 N. Iorgov
The corresponding eigenvectors |γ1, γ2, . . . , γn〉 are not standard basis vectors of course. To
obtain their coordinates in the standard basis we will use an iterative procedure as was promised
in the introduction. We start from the eigenvectors of open 1-particle BSQC. Then we construct
eigenvectors of open 2-particle BSQC by addition in an appropriate way one more particle. And
so on. In parallel with this procedure we have an iterative procedure of obtaining the amplitudes:
(κ11 := κ1)
+2nd particle−→ (κ21, κ22)
+3rd particle−→ · · · +nth particle−→ (κn1, . . . , κnn).
Now we will describe the procedure how to find these amplitudes κm,s, m = 1, 2, . . . , n, s =
1, 2, . . . ,m. We will need the variables
xm,s
m′,s′ =
κm,s
κm′,s′
, xm =
cm
dm
, xm,s =
cmκm
dmκm,s
, x̃m,s =
bmκm,s
amκm
, (6)
and variables ym,s
m′,s′ , ym, ym,s, ỹm,s. The latter are defined (up to a root of 1, which will be fixed
later) by condition that points pm,s
m′,s′ = (xm,s
m′,s′ , y
m,s
m′,s′), pm = (xm, ym), pm,s = (xm,s, ym,s), p̃ =
(x̃m,s, ỹm,s) belong to Fermat curve xN + yN = 1. First, we define κ1,1 := κ1. If we constructed
all the above variables for m − 1 particles, we define the variables κm,1, κm,2, . . . , κm,m by the
equations
κm,1κm,2 · · ·κm,m = κm−1,1κm−1,2 · · ·κm−1,m−1κm, (7)
κm
am−1dm
ym−1
ymym−1,lỹm−1,l
∏
s 6=l
ym−1,s
m−1,l
ym−1,l
m−1,s
m∏
k=1
ym−1,l
m,k
m−2∏
s=1
ym−2,s
m−1,l
= 1, l = 1, 2, . . . ,m− 1. (8)
We would like to mention that this iterative procedure has a similarity to iterative procedures
in [15, 16]. To solve these equations we first take N -th power of them. It gives us system of
linear equations
κN
m,1κN
m,2 · · ·κN
m,m = κN
m−1,1κN
m−1,2 · · ·κN
m−1,m−1κN
m , (9)
κN
m−1,l
aN
m−1d
N
m
yN
m−1
yN
myN
m−1,lỹ
N
m−1,l
m∏
k=1
(
1− κN
m,k/κN
m−1,l
)
m−2∏
s=1
(
1− κN
m−2,s/κN
m−1,l
) = 1, l = 1, 2, . . . ,m− 1. (10)
with respect to elementary symmetric polynomials in variables {κN
m,1, . . . , κN
m,m}. Solving equa-
tion with coefficients being the values of the mentioned symmetric polynomials we obtain the
values of {κN
m,1, . . . , κN
m,m}. The variables {κm,1, . . . , κm,m} can be found up to N -th roots of 1.
We fix their phases in a way to satisfy (7) and (8).
To compare these formulas with the formulas for eigenvalues proposed by Tarasov [14] we
consider polynomials Am(λN ) with zeroes ε/κN
m,s, s = 1, 2, . . . , m, where ε = (−1)N :
Am
(
λN
)
=
m∏
s=1
(
1− ε κN
m,sλ
N
)
, m ≥ 2; A1
(
λN
)
= 1− εκN
1 λN ; A0
(
λN
)
= 1. (11)
Then the relations (9) and (10) can be rewritten compactly as recursion relation for Am(λN ),
m ≥ 2:
Am
(
λN
)
=
((
1− εκN
mλN
)
+
cN
m − dN
m
cN
m−1 − dN
m−1
(
bN
m−1d
N
m−1
κN
m−1
− ελNaN
m−1c
N
m−1
))
Am−1
(
λN
)
Eigenvectors of Open Bazhanov–Stroganov Quantum Chain 5
+
cN
m−dN
m
cN
m−1−dN
m−1
(
bN
m−1−ελNaN
m−1κN
m−1
)(
ελNcN
m−1κN
m−1−dN
m−1
)
κN
m−1
Am−2
(
λN
)
.(12)
Indeed, the relation (9) follows from the relation for coefficients in (12) at (λN )m. If we fix
sequentially λN = ε/κN
m−1,l, l = 1,2, . . . ,m − 1, (that is by the zeroes of Am−1(λN )) we obtain
the relations (10).
This recursion relation for Am(λN ) can be obtained by means of averaged L-operators [14].
Using
Lm(λN ) =
(
1− εκN
mλN −ελN
(
aN
m − bN
m
)
cN
m − dN
m bN
mdN
m/κN
m − ελNaN
mcN
m
)
we define polynomials Am(λN ), Bm(λN ), Cm(λN ) and Dm(λN ) by(
Am
(
λN
)
Bm
(
λN
)
Cm
(
λN
)
Dm
(
λN
) ) = L1
(
λN
)
L2
(
λN
)
· · · Lm
(
λN
)
. (13)
In particular, we have
Am
(
λN
)
=
(
1− εκN
mλN
)
Am−1
(
λN
)
+
(
cN
m − dN
m
)
Bm−1
(
λN
)
,
Bm
(
λN
)
= −ελN
(
aN
m − bN
m
)
Am−1
(
λN
)
+
(
bN
mdN
m/κN
m − ελNaN
mcN
m
)
Bm−1
(
λN
)
. (14)
Excluding Bm−1(λN ) from these two relations we get
Bm
(
λN
)
=
bN
mdN
m/κN
m − ελNaN
mcN
m
cN
m − dN
m
Am
(
λN
)
−
detLm
(
λN
)
cN
m − dN
m
Am−1
(
λN
)
.
Substituting the right-hand side of this equation with m replaced by m−1 instead of Bm−1(λN )
in (14) we get (12). Therefore two formulas (12) and (13) for Am(λN ) are equivalent. Summa-
rizing, in order to find amplitudes κm,s, s = 1, . . . ,m, for some m, we have to find Am(λN )
using (12) or (13). Then solving equation Am(λN ) = 0 of mth degree with respect to λN and
taking into account (11) we can find κN
m,s, s = 1, . . . ,m. This gives us the set κm,s up to Nth
roots of 1. At last step, we have to fix their values in a way to satisfy (7) and (8).
It seems to the author that the equation Am(λN ) = 0 can not be solved explicitly in the
case of generic parameters. In the next section, the solution for the homogeneous RTC is given
explicitly. The author does not know other interesting special cases of parameters which admit
explicit solution for the spectrum of Am(λ). As shown in [11], it is possible to give an explicit
solution for the spectrum of Bm(λ) in the homogeneous case of m-particle Bazhanov–Stroganov
quantum chain.
4 Amplitudes for the homogeneous Relativistic Toda Chain
In this section we sketch the method described in [15] of obtaining the amplitudes for the
homogeneous RTC: ak = a, bk = 0, ck = 0, dk = d, κk = κ. In this case the amplitudes κm,s,
s = 1, . . . ,m, can be expressed in terms of solutions of some quadratic equation. Since
LRTC
k
(
λN
)
= LRTC(λN ) =
(
1− εκNλN −εaNλN
−dN 0
)
, (15)
we obtain(
Am
(
λN
)
Bm
(
λN
)
Cm
(
λN
)
Dm
(
λN
) ) =
(
LRTC
(
λN
))m
.
6 N. Iorgov
Applying the fact that 2× 2 matrix M with eigenvalues µ+ and µ− satisfies
Mm =
µm
+ − µm
−
µ+ − µ−
M −
µm
+µ− − µm
−µ+
µ+ − µ−
1
for matrix LRTC(λN ) we obtain
Am
(
λN
)
=
(
1− εκNλN
)xm
+ − xm
−
x+ − x−
−
xm
+x− − xm
−x+
x+ − x−
, (16)
where x+(λN ) and x−(λN ) are eigenvalues of L(λN ). These eigenvalues are roots of characteristic
polynomial x2 − τ(λN )x + δ(λN ) = 0:
x± =
1
2
(
τ ±
√
τ2 − 4δ
)
,
where, using (15),
τ
(
λN
)
= trLRTC
(
λN
)
= x1 + x2 = 1− εκNλN , (17)
δ
(
λN
)
= detLRTC
(
λN
)
= x1x2 = −εaNdNλN . (18)
Taking into account (17) we rewrite (16) as
Am
(
λN
)
=
xm+1
+ − xm+1
−
x+ − x−
.
Introducing the variable φ by x+/x− = eiφ we find that roots of Am correspond to roots φm,s
of ei(m+1)φ = 1 (without φ = 0) that is
φm,s = 2πs/(m + 1), s = 1, 2, . . . ,m. (19)
Now we need to find an explicit relation between λN and φ. We have
τ +
√
τ2 − 4δ = eiφ
(
τ −
√
τ2 − 4δ
)
.
Therefore
τ2 = 4δ cos2
φ
2
. (20)
Taking into account (17) and (18) we consider (20) as quadratic equation with respect to λN :
λ2Nκ2N + 2ελN
(
aNdN + aNdN cos φ− κN
)
+ 1 = 0.
The solution λN (φ) of this equation describes the relation between the variables λN and φ.
Therefore we can translate the zeroes (19) ofAm(λN (φ)) in terms of variable φ to zeroes λN (φm,s)
in terms of λN . Finally, taking into account (11) we find
κN
m,s = ε/λN (φm,s), s = 1, 2, . . . ,m.
Eigenvectors of Open Bazhanov–Stroganov Quantum Chain 7
5 Eigenvectors and eigenvalues
In order to give explicit formulas for the eigenvectors of An(λ) we remind the definition (see
for example [17]) of wp(s) which is an analogue of Γq-function at root of unity. For any point
p = (x, y) of Fermat curve xN + yN = 1, we define wp(s), s ∈ ZN , by
wp(s)
wp(s− 1)
=
y
1− xωs
, wp(0) = 1. (21)
The function wp(s) is cyclic: wp(s + N) = wp(s).
We will use the notation |γn〉 ∈ V1 ⊗ · · · ⊗ Vn for eigenvectors of the operator An(λ) of
the BSQC with n particles. These eigenvectors are labeled by n parameters γn,s ∈ ZN , s =
1, 2, . . . , n, collected into a vector
γn = (γn,1, . . . , γn,n) ∈ (ZN )n.
The following theorem gives a procedure of obtaining the eigenvectors |γn〉 of An(λ) from
the eigenvectors |γn−1〉 ∈ V1 ⊗ · · · ⊗ Vn−1 of An−1(λ) and basis vectors |γn〉n ∈ Vn. To find
the formula for |γn−1〉 we can use the same theorem and so on. At the last step we need the
eigenvectors of 1-particle quantum chain.
From (3) and (4), it is easy to see that the vectors |γ1,1〉1 ∈ V1, γ1,1 ∈ ZN , are eigenvectors
for A1(λ):
A1(λ)|γ1,1〉1 = (1 + κ1,1ω
γ1,1) |γ1,1〉1,
where κ1,1 = κ1.
In what follows, the vector γ±k
n means the vector γn in which γn,k is replaced by γn,k ± 1.
Theorem 1. The vector |γn〉 = |γn1, . . . , γnn〉
|γn〉 =
∑
γn−1∈(ZN )n−1
Q(γn−1|γn)|γn−1〉 ⊗ |σn〉n (22)
satisfies
An(λ)|γn〉 =
n∏
k=1
(1 + κn,kω
γn,kλ) |γn〉 =
n∏
k=1
(1− λ/λn,k) |γn〉, (23)
Bn (λn,k) |γn〉 =
anλn,k
yn
(
1− xn,kω
−γn,k−1
)(
1− x̃n,kω
γn,k
)(n−1∏
s=1
yn−1,s
n,k
)
|γ+k
n 〉, (24)
Bn(λ)|γn〉 = λ
an
yn
n∑
k=1
∏
s 6=k
λ− λn,s
λn,k − λn,s
×
(
1− xn,kω
−γn,k−1
)(
1− x̃n,kω
γn,k
)(n−1∏
l=1
yn−1,l
n,k
)
|γ+k
n 〉, n > 1, (25)
B1(λ)|γ1〉 = λa1
(
1− x̃1,1ω
γ1,1
)
|γ+1
1 〉, (26)
if |γn−1〉 = |γn−1,1, . . . , γn−1,n−1〉 ∈ V1 ⊗ · · · ⊗ Vn−1 satisfies the same relations with n replaced
by n− 1. In the above formulas we used
Q(γn−1|γn) =
ωγn−1,1+···+γn−1,n−1
n−1∏
l=1
n∏
k=1
w
pn−1,l
n,k
(γn−1,l − γn,k)
wpn(−σn − 1)
n−1∏
j,l=1
(j 6=l)
w
pn−1,j
n−1,l
(γn−1,j − γn−1,l)
·
n−1∏
l=1
wpn−1,l
(−γn−1,l − 1)
n−1∏
l=1
wp̃n−1,l
(γn−1,l − 1)
,
8 N. Iorgov
Q(γ1|γ2) =
ωγ1,1w
p1,1
2,1
(γ1,1 − γ2,1)wp1,1
2,2
(γ1,1 − γ2,2)
wp2(γ1,1 − γ2,1 − γ2,2 − 1)wp̃1,1(γ1,1 − 1)
,
λm,s = −ω−γm,s/κm,s, σn(γn−1,γn) ≡ σn =
n∑
k=1
γn,k −
n−1∑
l=1
γn−1,l. (27)
Proof. We suppose that the formulas (23) and (25) with n replaced by n − 1 are proved. To
prove the action formulas (23) and (25) we use the recurrent relations
An(λ) = An−1(λ) (1 + λκnvn) + Bn−1(λ) un(cn − dnvn), (28)
Bn(λ) = An−1(λ) λu−1
n (an − bnvn) + Bn−1(λ)
(
λancn +
bndn
κn
vn
)
(29)
which follow from (4).
The action formula for An(λ): To prove the action formula (23) we act by both sides of (28)
on (22) and use the formulas (23) and (25) with n replaced by n − 1. After shifting in an
appropriate way the variables of summation γn−1 we reduce the problem to verification of
relation(
n∏
k=1
(1− λ/λn,k)−
n−1∏
l=1
(1− λ/λn−1,l) (1 + λκnωσn)
)
Q(γn−1|γn)
= λ
an−1
yn−1
n−1∑
l=1
n−1∏
s=1
s 6=l
λ− λn−1,s
ωλn−1,l − λn−1,s
(1− xn−1,lω
−γn−1,l
)(
1− x̃n−1,lω
γn−1,l−1
)
×
(
n−2∏
s=1
yn−2,s
n−1,l
)(
cn − dnωσn+1
)
Q(γ−l
n−1|γn). (30)
Using
Q(γ−l
n−1|γn)
Q(γn−1|γn)
an−1
yn−1
(1− xn−1,lω
−γn−1,l)(1− x̃n−1,lω
γn−1,l−1)
(
n−2∏
s=1
yn−2,s
n−1,l
)
=
κn
ωdn
n∏
k=1
(
1− λn,k
λn−1,l
)
1− xnω−σn−1
∏
s 6=l
1− ω
λn−1,l
λn−1,s
1− λn−1,s
λn−1,l
which follows directly from (27) and (8), we rewrite (30) as
n∏
k=1
(1− λ/λn,k)−
n−1∏
l=1
(1− λ/λn−1,l) (1 + λκnωσn)
= λ
n−1∑
l=1
n−1∏
s=1
s 6=l
λ− λn−1,s
λn−1,l − λn−1,s
−κnωσn
λn−1,l
n∏
k=1
λn,k
n−1∏
s=1
λn−1,s
n∏
k=1
(1− λn−1,l/λn,k) .
Taking into account
−κnωσn
n∏
k=1
λn,k
n−1∏
s=1
λn−1,s
= 1, (31)
Eigenvectors of Open Bazhanov–Stroganov Quantum Chain 9
which follows from (7), we obtain finally
n∏
k=1
(1− λ/λn,k)−
n−1∏
l=1
(1− λ/λn−1,l) (1 + λκnωσn)
= λ
n−1∑
l=1
n−1∏
s=1
s 6=l
λ− λn−1,s
λn−1,l − λn−1,s
1
λn−1,l
n∏
k=1
(1− λn−1,l/λn,k) .
To verify this equality we note that both sides are polynomials in λ of degree n− 1 (not n due
to (31)) without free term. Therefore it is sufficient to verify this relation at n − 1 different
values of λ. Taking these values to be λ = λn−1,l, l = 1, 2, . . . , n−1, we easily prove the relation.
Thus we proved (23).
The action formula for Bn(λn,k): Next we show the validity of (24). The action formulas
for B1(λ) and B2(λ) can be verified in a direct way. Thus we suppose n > 2.
Excluding Bn−1(λ) from (28) and substituting it into (29) we get
un(cn − dnvn)Bn(λ) =
(
λancn + ω
bndn
κn
vn
)
An(λ)
− ωλandnvn
(
1 +
λcnκn
ωdn
)(
1 +
bn
λanκn
)
An−1(λ). (32)
Let us apply (32) to |γn〉 for λ = λn,k = −ω−γn,k/κn,k, i.e. at the zeros of eigenvalue of An(λ).
This gives, by virtue of the definitions (6) of xm,s and x̃m,s:
un(cn − dnvn)Bn(λn,k)|γn〉 = −λn,kωandn
(
1− xn,kω
−γn,k−1
)(
1− x̃n,kω
γn,k
)
×
∑
γn−1∈(ZN )n−1
ωσnQ(γn−1|γn)An−1(λn,k)|γn−1〉 ⊗ |σn〉. (33)
From (23) we know how to apply An−1 to |γn−1〉:
An−1(λn,k)|γn−1〉 =
n−1∏
s=1
(
1− κn−1,s
κn,k
ωγn,k−γn−1,s
)
|γn−1〉. (34)
Using (2) we find the action of the inverse of the operator un(cn − dnvn) on |σn〉n :
(un(cn − dnvn))−1|σn〉n =
(
cn − dnωσn+1
)−1|σn + 1〉n. (35)
Taking into account (27) and (21) we get
Q(γn−1|γn)
ωσn
n−1∏
s=1
(
1− κn−1,s
κn,k
ωγn,k−γn−1,s
)
cn − dnωσn+1
= −ω−1 Q(γn−1|γ+k
n )
n−1∏
s=1
yn−1,s
n,k
dnyn
. (36)
Finally, using (34), (35) and (36), we reduce (33) to (24).
The action formula for Bn(λ): From (3) and (4) it is easy to find that the operator Bn(λ)/λ
is a polynomial in λ of (n − 1)th order. Due to (24) we know the action formulas for Bn(λ)/λ
at the n particular values of λ: λ = λn,k, k = 1, 2, . . . , n. This data is enough to reconstruct
the action of the polynomial Bn(λ) on |γn〉 uniquely. Lagrange interpolation formula gives
Bn(λ)
λ
|γn〉 =
n∑
k=1
∏
l 6=k
λ− λn,l
λn,k − λn,l
Bn(λn,k)
λn,k
|γn〉.
Finally using (24) we get (25). This completes the proof of the Theorem. �
10 N. Iorgov
6 Discussion
In this contribution we applied the iterative procedure of obtaining the eigenvectors for quantum
integrable systems by Kharchev and Lebedev [7] (which has origin in [8] by Sklyanin) to open
Bazhanov–Stroganov quantum chain. We plan to extend this result (along the line of the
paper [10]) to the case of Bazhanov–Stroganov chain with integrable boundary interaction.
Acknowledgements
The author would like to acknowledge the organizers of the Sixth International Conference
“Symmetry in Nonlinear Mathematical Physics” (June 20–26, Kyiv) for their nice conference.
The present paper is the written version of the talk delivered by the author at this conference.
The author is thankful to Professors G. von Gehlen, S. Pakuliak and V. Shadura for collaboration
in obtaining the results presented in this contribution. The research presented here is partially
supported by INTAS (grant No.03-51-3350) and by the French–Ukrainian project “Dnipro”.
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1 Introduction
2 Bazhanov-Stroganov quantum chain
3 Eigenvalues and associated amplitudes
4 Amplitudes for the homogeneous Relativistic Toda Chain
5 Eigenvectors and eigenvalues
6 Discussion
|