On B¨ acklund transformations and boundary conditions associated with the quantum inverse problem for a discrete nonlinear
arXiv:math-ph/0202027v1 19 Feb 2002
integrable system and its connection to Baxter’s Q-operator
A. Ghose Choudhury, Department of Physics, Surendranath College, 24/2 Mahatma Gandhi Road, Calcutta-700 009, India email:
[email protected] and A. Roy Chowdhury High Energy Physics Division Department of Physics, Jadavpur University Calcutta -700 032, India. email:
[email protected]
Abstract A discrete nonlinear system is analysed in case of open chain boundary conditions at the ends. It is shown that the integrability of the system remains intact, by obtaining a modified set of Lax equations which automatically take care of the boundary conditions. The same Lax pair also conforms to the conditions stipulated by Sklyanin [5]. The quantum inverse problem is set up and the diagonalisation is carried out by the method of sparation of variables. B¨acklund transformations are then derived under the modified boundary conditions using the classical r-matrix . Finally by quantising the B¨acklund transformation it is possible to identify the relation satisfied by the eigenvalue of Baxter’s Q-operator even for the quasi periodic situation.
1
Introduction
Analysis of nonlinear integrable systems is a subject of immense importance both from the physical and mathematical points of view. The inverse problem formulated on the basis of Lax operators can solve a wide class of problems. On the other hand, it is also important that problems be formulated with pre-assigned boundary conditions and solutions of the corresponding Cauchy problem be obtained. However at times the imposition of finite boundary conditions may lead to a loss of integrability. An important developement in this regard was the discovery by Sklyanin [1], who showed how non-periodic boundary conditions could be imposed on an integrable model without destroying its integrability. In this communication we have analysed a discrete nonlinear system, (also known as the DST model), initially studied by Ragnisco et al and also recently by Sklyanin et al from the view point of its integrability in presence of open boundary conditions. Our approach is different from that of Sklyanin in the sense that we have changed the form of the Lax operator at the end points where the boundary conditions are imposed. We have then shown that our approach can also accomodate the conditions laid down by Sklyanin for the system to be integrable under such circumstances. The present formalism was first used by Zhou [3] for a fermionic spin system. Both the classical and quantum hamiltonians have been derived explicitly. The quantum R matrix has also been deduced so as to formulate the quantum inverse problem for the model under finite boundary conditions. Diagonalisation of the hamiltonian is carried out by using the method of separation of variables, as the boundary matrices are non diagonal in character. Of late there has been a great deal of interest in the study of canonical B´acklund transformations within the framework of classical r matrix theory, specially under periodic boundary conditions [15]. We have discussed this issue in case of quasi -periodic and finite boundary conditions. Finally in the case of the less stringent quasi-periodic boundary conditions we have shown how a connection with the eigenvalue of Baxter’s Q operator may be established [11] by quantising the B¨acklund transformation. The model under analysis is described by the following equations of motion. q˙n = qn+1 − qn2 rn r˙n = −rn−1 + qn rn2 1
(1.1)
where q˙n stands for the time derivative of qn . It was originally proposed by Ragnisco et al [2] in their analysis of Lie-B¨acklund symmetries of discrete systems. The Lax pair associated with (1.1) may be written as Ψn+1 = Ln Ψn where
Ln (λ) =
Ψnt = Mn Ψn
λ + qn rn qn rn
1
Mn (λ) =
(1.2)
λ 2
qn
rn−1 − λ2
(1.3)
Note that consistency of (1.2) yields the equations of motion only for periodic boundary conditions. If however certain nontrivial boundary conditions are to be introduced then one has to adopt a different strategy as will be explained in the sequel. The system given by (1.1) possesses a hamiltonian structure with the following symplectic form [4].
qn rn
t
=
0
δH δqn
δH δrn
1
−1 0
H=
1X (qn+1 rn + qn rn−1 − qn2 rn2 ) 2 n
(1.4)
For the purpose of introducing the boundary conditions we modify the Lax operator in (1.2) in the following manner, Ψj+1 = Lj (λ)Ψj ,
j = 1, 2, ....N
dΨj = Mj (λ)Ψj j = 2, 3, ....N − 1 dt dΨN +1 = WN +1 (λ)ΨN +1 dt dΨ1 = W1 (λ)Ψ1 dt
(1.5)
where WN +1 , W1 are two new 2 × 2 matrices depending on the spectral parameter λ and on the dynamical variables. The usual consistency condition of (1.2) viz dLj (λ) = Mj+1 Lj (λ) − Lj (λ)Mj (λ) dt is now replaced by the following set of equations: dLj (λ) = Mj+1 (λ)Lj (λ) − Lj (λ)Mj (λ) (j = 1, 2....N − 1) dt dLN (λ) = WN +1 LN (λ) − LN (λ)MN (λ) dt 2
(1.6)
dL1 (λ) = M2 (λ)L1 (λ) − L1 (λ)W1 (λ) (1.7) dt That is the consistency conditions at the two ends are different. In the present case we have formally.
WN +1 =
λ 2
rN
qN +1 − λ2
W1 =
λ 2
q1
r0 − λ2
(1.8)
so that upon imposing the boundary conditions r0 = θ− , qN +1 = θ+ the equations for the two ends of the discrete chain turn out to be as follows. q˙1 = q2 − q12 r1 2 q˙N = (θ+ − qN rN )
r˙1 = −(θ− − q1 r12 ) 2 r˙N = −(rN −1 − qN rN )
(1.9)
We shall refer to the above system as an open chain.
2
Boundary conditions and classical r− matrix
To ascertain the hamiltonian associated with the open chain system it is always useful to compute the so called classical r-matrix through the poisson bracket given by the symplectic form in (1.4) [8]. It is straight forward to show that. {Ln (λ) ⊗, Lm (µ)} = [r(λ, µ), Ln (λ) ⊗ Ln (µ)]δnm
(2.1)
P , P standing for the permutation operator. The mondromy matrix is where r(λ, µ) = − λ−µ
then defined by TN (λ) =
N Y
Ln (λ)
(2.2)
n=1
From (2.1) and (2.2) it follows that {TN (λ) ⊗, TN (µ)} = [r(λ, µ), TN (λ) ⊗ TN (µ)]
(2.3)
We shall refer to this poisson algebra as the classical inverse scattering method (CISM-I) algebra. It follows that N X dTN LN ...Ln+1 L˙ n Ln−1 ....L1 (λ) = WN +1 TN (λ) − TN (λ)W1 (λ) = dt n=1
(2.4)
from which it is easy to conclude that J(λ) = trTN (λ) is a constant of motion in the periodic case i.e when qm+N = qm and rm+N = rm . 3
2.1
Quasiperiodic case
Instead of the periodic boundary conditions that are customarily assumed, one can impose the so called quasiperiodic boundary conditions on the system, without destroying its integrability. This requires the existence of a matrix C(λ, ξ) such that ¯ J(λ) = tr[C(λ, ξ)TN (λ)]
(2.5)
is time independent so that in the present situation one has ¯ dJ(λ) = tr[C(λ, ξ)(MN +1TN − TN M1 )] = tr[C(λ, ξ)MN +1 TN ] − tr[M1 C(λ, ξ)TN ] dt
(2.6)
from which it follows that C(λ, ξ)MN +1 = M1 C(λ, ξ)
(2.7)
As a specific example one can take
C(ξ) =
ξ
− 21
0
0 1
ξ2
(2.8)
¯ To analyse the structure of J(λ) in detail, we note the following asymptotics of the monodromy matrix in our case. TN11 (λ) ≈ λN + λN −1 S + p2 λN −2 + .... TN12 (λ) ≈ λN −1 q1 + O(λN −2) TN21 (λ) ≈ λN −1 rN + O(λN −2) TN22 ≈ λN −2 p′2 + O(λN −2 )
(2.9)
whence we get 1 1 1 1 J¯(λ) = tr(C(ξ)TN (λ)) = ξ − 2 λN + (ξ − 2 S)λN −1 + (ξ − 2 p2 )λN −2 + (ξ 2 p′2 )λN −2 + .... (2.10)
where S=
N X i=1
si , p2 =
N −1 X
(qi+1 ri ) +
X
si sj ,
p′2 = rN q1
(2.11)
i
