Wronskian-type formula for inhomogeneous $$TQ$$ equations
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WRONSKIAN-TYPE FORMULA FOR INHOMOGENEOUS T Q EQUATIONS Rafael I. Nepomechie∗
It is known that the transfer-matrix eigenvalues of the isotropic open Heisenberg quantum spin-1/2 chain with nondiagonal boundary magnetic fields satisfy a T Q equation with an inhomogeneous term. We derive a discrete Wronskian-type formula relating a solution of this inhomogeneous T Q equation to the corresponding solution of a dual inhomogeneous T Q equation.
Keywords: Bethe ansatz, T Q equation, discrete Wronskian, boundary integrability DOI: 10.1134/S004057792009007X
1. Introduction and summary of results We consider the famous Baxter T Q equation for the closed periodic XXX spin chain of length N : T (u)Q(u) = (u+ )N Q−− (u) + (u− )N Q++ (u).
(1.1)
Here and hereafter, we use the brief notation f ± (u) = f (u ± i/2) and f ±± (u) = f (u ± i). It is known that for a given transfer-matrix eigenvalue T (u), Eq. (1.1) can be regarded as a second-order finite-difference equation for Q(u). The eigenvalue T (u) is necessarily a polynomial in u (of degree N ) because the model is integrable. It is well known that Eq. (1.1) has two independent polynomial solutions [1]. One of them is a polynomial Q(u) of degree M ≤ N/2 of the form Q(u) =
M
(u − uk ),
(1.2)
k=1
whose zeros {uk } are solutions of the Bethe equations that follow directly from (1.1)
uj + i/2 uj − i/2
N =
M uj − uk + i , uj − uk − i
j = 1, . . . , M.
(1.3)
k=1, k=j
The other is a polynomial P (u) of degree N − M + 1 > N/2 corresponding to Bethe roots “on the other side of the equator.” These two solutions are related by a discrete Wronskian (or Casoratian) formula P + (u)Q− (u) − P − (u)Q+ (u) ∝ uN ,
(1.4)
∗
Physics Department, University of Miami, Coral Gables, Florida, USA, e-mail: [email protected]. This research was supported in part by a Cooper fellowship. Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 3, pp. 430–435, September, 2020. Received January 17, 2020. Revised March 23, 2020. Accepted March 23, 2020. c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2043-1195
1195
where ∝ denotes equality up to a multiplicative constant. The existence of a second polynomial solution of the T Q equation is equivalent to the admissibility of the Bethe roots [2], [3]. Using the Wronskian formula, we can succinctly reformulate the Q-system for this model [4] (which provides an efficient way to compute the admissible Bethe roots) in terms of Q and P [5], [6]. A generalization of Wronskian formula (1.4) for the open XXX spin chain with diagonal boundary fields was recently obtained [7]: g(u)P + (u)Q− (u) − f (u)P − (u)Q+ (u) ∝ u2N +1 ,
(1.5)
where Q(u) and P (u) are the respective polynomial solutions of a T Q and a dual T Q equation (see Eqs. (2.7) and (2.12) below). Moreover, the functions f (u) and g(u) are given by (diagonal case) f (u) = (u − iα)(u + iβ),
g(u) = f (−u) = (u + iα)(u − iβ),
(1.6)
where α and β are boundary parameters. This result was used in [7]
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