Spectral analysis of an open $q$-difference Toda chain with two-sided boundary interactions on the finite integer lattice

A quantum $n$-particle model consisting of an open $q$-difference Toda chain with two-sided boundary interactions is placed on a finite integer lattice. The spectrum and eigenbasis are computed by establishing the equivalence with a previously studied $q$-boson model from which the quantum integrability is inherited. Specifically, the $q$-boson-Toda correspondence in question yields Bethe Ansatz eigenfunctions in terms of hyperoctahedral Hall-Littlewood polynomials and provides the pertinent solutions of the Bethe Ansatz equations via the global minima of corresponding Yang-Yang type Morse functions.


Introduction
The relativistic Toda chain is an ubiquitous one-dimensional n-particle model introduced by Ruijsenaars that is integrable both at the level of classical and quantum mechanics [R90].In the case of an open chain, integrable perturbations at the boundary were implemented via the boundary Yang-Baxter equation [KT96,S90].At the quantum level, the hamiltonian of the relativistic Toda chain is given by a (q-)difference operator.Quantum groups connect the difference operator at issue to the quantum K-theory of flag manifolds [BF05,GL03] and provide a natural representation-theoretical habitat for the construction of its eigenfunctions [E99, S00].
When considering the quantum dynamics on an integer lattice the eigenvalue problem for the q-difference Toda chain can be solved in terms of q-Whittaker functions that arise as a parameter specialization of the Macdonald polynomials, both in the case of particles moving on an infinite lattice [GLO11] and in the case of particles moving on a finite periodic lattice [DP16].From the perspective of integrable probability, such particle models are of interest in connection with the q-Whittaker process [BC14].Representation-theoretical constructions for the pertinent q-Whittaker functions can be found in [DK18,DKT17,FJM09].
This note addresses the spectral problem for an open n-particle q-difference Toda chain on the finite lattice {0, 1, 2, . . ., m} that is endowed with two-parameter boundary interactions on both ends.The model could be thought of as a finite discrete and q-deformed counterpart of Sklyanin's open quantum Toda chain with general two-sided boundary perturbations governed by Morse potentials [S88].In the limit m → ∞ the pertinent q-difference Toda hamiltonian was diagonalized in terms of hyperoctahedral q-Whittaker functions that arise in turn through a parameter specialization of the Macdonald-Koornwinder polynomials [DE15].Here it will be shown that for finite m an explicit eigenbasis can be constructed from Bethe Ansatz wave functions given by Macdonald's hyperoctahedral Hall-Littlewood polynomials [M00].The main idea is to exploit an equivalence between q-difference Toda chains and q-boson models pointed out in [DP16].By establishing a version of this equivalence in the current situation of an open chain with boundary perturbations, our q-difference Toda hamiltonian is mapped to the hamiltonian of a q-boson model previously diagonalized in [DEZ18].The upshot is that the commuting quantum integrals and the Bethe Ansatz eigenfunctions for the q-difference Toda chain can in this approach be retrieved directly from those in [DEZ18] for the corresponding q-boson model.
The material is organized as follows.Section 2 describes the hamiltonian of our q-difference Toda chain and verifies its self-adjointness.Section 3 establishes the equivalence with the q-boson model from [DEZ18] and therewith retrieves the corresponding Bethe Ansatz wave functions in terms of hyperoctahedral Hall-Littlewood polynomials.The Bethe Ansatz equations of interest are of a convex type studied in wider generality in [DE19], which entails an explicit description of the spectrum via the global minima of associated Yang-Yang-type Morse functions detailed in Section 4. The presentation closes in Section 5 with a description of the spectral analysis for the q-difference Toda chain in the degenerate limit q → 1.
2. Open q-difference Toda chain with boundary interactions 2.1.Quantum hamiltonian.Given m, n ∈ N, the q-difference Toda chain under consideration describes the quantum dynamics of n interacting particles hopping over the finite integer lattice {0, 1, 2, . . ., m}.The positions of these particles are encoded by a partition µ = (µ 1 , µ 2 , . . ., µ n ) in the configuration space (2.1) The dynamics is governed in turn by the following quantum hamiltonian Here T i and T −1 i denote hopping operators that act on n-particle wave functions ψ via a unit translation of the ith particle to the left and to the right, respectively: The action of H (2.2) on wave functions ψ : Λ (n,m) → C is well-defined in the sense that the coefficient of (T ǫ i ψ)(µ 1 , . . ., µ n ) in (Hψ)(µ 1 , . . ., µ n ) vanishes for any (µ 1 , . . ., µ n ) ∈ Λ (n,m) such that (µ 1 , . . ., µ i−1 , µ i + ǫ, µ i+1 , . . ., µ n ) ∈ Λ (n,m) .Notice also that the convention in the second brace below Eq. (2.2) can be interpreted as representing the positions of two additional particles fixed at the lattice end-points 0 and m, respectively.The parameter q ∈ (−1, 1) \ {0} denotes a scale parameter of the model governing the nearest neighbour interaction between the particles whereas the parameters α ± ∈ (−1, 1) and β ± ∈ R represent coupling constants regulating additional interactions at the boundary of the chain.
Remark 2.2.For m → ∞, the q-difference Toda hamiltonian H (2.2) was diagonalized in [DE15,Section 7] in terms of a unitary eigenfunction transform with a q-Whittaker kernel built from a parameter specialization of the Macdonald-Koornwinder polynomials.
2.2.Proof of Proposition 2.1.The action of where the vectors e 1 , . . ., e n represent the standard unit basis for Z n .

Eigenfunctions
3.1.Bethe Ansatz.While Proposition 2.1 implies that the existence of an orthogonal eigenbasis diagonalizing H (2.2) in ℓ 2 Λ (n,m) , ∆ is evident from the spectral theorem for self-adjoint operators in finite dimension, the aim here is to provide an explicit eigenbasis given by Bethe Ansatz wave functions in the spirit of [D21] for n = 1.
From now on it will moreover be assumed (unless explicitly stated otherwise) that the boundary parameters α ± , β ± have values such that the roots p ± , q ± of the two quadratic polynomials x 2 − β ± x + α ± belong to the interval (−1, 1) \ {0}: m) , ∆ through its values on Λ (n,m) as follows: m) ). (3.5a) The wave function ψ ξ (3.5a) solves the eigenvalue equation for the q-difference Toda hamiltonian H (2.2) provided the spectral parameter ξ ∈ R m reg satisfies the algebraic system of Bethe Ansatz equations ,n) is in fact a symmetric polynomial in cos(ξ 1 ), . . ., cos(ξ m ) of total degree Hence, it is clear that the Bethe Ansatz wave function ψ ξ in Theorem 3.1 extends smoothly in the spectral parameter ξ from values in R m reg to values in R m .

Spectral analysis
4.1.Solutions for the Bethe Ansatz equations.The following system of transcendental equations provides a logarithmic form of the Bethe Ansatz equations in Theorem 3.1: Indeed, upon multiplying Eq. (4.1a) by i (= √ −1) and applying the exponential function on both sides it is readily seen that any of its solutions gives rise to a solution of the Bethe Ansatz equations (3.5c) (where-recall-the boundary parameters α ± , β ± and p ± , q ± are related via Eq.(3.4a)).
For any κ ∈ Λ (m,n) , the system in Eqs.(4.1a), (4.1b) describes the critical point of a Yang-Yang type Morse function: The function V κ (ξ 1 , . . ., ξ m ) belongs to a wider class of smooth, strictly convex and radially unbounded Morse functions studied in [DE19, Section 3].The upshot is that via a Yang-Yang type analysis, one arrives at m+n n solutions for the Bethe Ansatz equations given by the respective minima of Proposition 4.1 (Solutions for the Bethe Ansatz Equations).Let q ∈ (−1, 1) \ {0} and let the boundary parameters α ± , β ± belong to the domain specified in Eqs.
(ii) The global minima ξ κ , κ ∈ Λ (m,n) in part (i) are all distinct and located within the open alcove Moreover, at a global minimum ξ = ξ κ the following estimates are fulfilled: where Proof.The assertions of this proposition follow by applying [DE19, Propositions 3.1 and 3.2] to the Bethe Ansatz equations of Theorem 3.1 (cf.[DEZ18, Remark 3.5]).
Remark 4.2.From a mostly academic perspective, Proposition 4.1 invites us to compute ξ κ via the gradient flow of the pertinent Morse function: which gives rise to the following system of differential equations (cf.Eq. (4.1a)) For n = 0 and κ = (0 m ), the corresponding gradient flow was analyzed in [D19] and seen to converge exponentially fast to the roots of the Askey-Wilson polynomial p m (cos ϑ; p + , q + p − , q − |q) [KLS10, Equation (14.1.1)]located within the interval of orthogonality 0 < ϑ < π.A minor variation of [D19, Theorem 2] reveals that in our present setting the equilibrium ξ κ of the gradient system in Eq. (4.5) remains globally exponentially stable, i.e. for any initial condition ξ κ (0) the unique solution ξ κ (t), t ≥ 0 of the gradient system converges exponentially fast to the equilibrium ξ κ .More specifically, by slightly adapting the analisis in [D19, Section 4] one readily deduces that for any 0 < ε < 2(n + k − ) (= a lower bound for the eigenvalues of the hessian of V κ (ξ 1 , . . ., ξ m )), there exists a constant ) where ξ ∞ ≡ max 1≤j≤m |ξ j |.Apart from ε, the actual value of the constant C ε in the uniform estimate of the error term will depend on the choice of the initial condition ξ κ (0) ∈ R m , as well as on κ ∈ Λ (m,n) and q, p ± , q ± ∈ (−1, 1) (cf.Remark 4.4 below).Notice that the q-difference Toda hamiltonian H (2.2) degenerates to a discrete Laplacian on Λ (n,m) in the symplectic Schur limit α ± , β ± , q → 0. At this elementary point in the parameters space, one has that (cf.Eq. (4.3b)) , . . ., π(1+κm) m+n+1 , which serves as a convenient initial condition for the gradient flow (4.5).Indeed, at this particular value of the spectral parameter the bounds in Eqs.(4.3b) and (4.3c) are fulfilled for any p ± , q ± , q ∈ (−1, 1).4.2.Spectrum and eigenbasis.By combining Theorem 3.1 and Proposition 4.1, an eigenbasis of Bethe Ansatz wave functions for the q-difference Toda hamiltonian H (2.2) in the Hilbert space ℓ(Λ (n,m) , ∆) is found together with the corresponding eigenvalues.(i) The spectrum of the q-difference Toda hamiltonian H (2.2) in the Hilbert space ℓ(Λ (n,m) , ∆) consists of the eigenvalues E(ξ κ ), κ ∈ Λ (m,n) , where E(ξ) is given by Eq. (3.5b).
Remark 4.6.In [DEZ18], Eq. (3.6) is interpreted as the eigenvalue equation for a hamiltonian of an m-particle q-boson model on the lattice {0, 1, . . ., n}.The quantum integrability of this m-particle q-boson hamiltonian, which is thus given explicitly by the difference operator acting at the RHS of Eq. (3.6), was established for α + = α − = 0 in [DE17] (using the quantum inverse scattering method) and for general boundary parameters in [DEZ18, Section 8] (using representations of the double affine Hecke algebra of type C ∨ C at the critical level q = 0).As detailed explicitly for the hamiltonian in the proof of Theorem 3.1, the mapping µ → µ ′ from Λ (n,m) onto Λ (m,n) allows us to pull back the commuting quantum integrals for the q-boson model from ℓ(Λ (m,n) , ∆ ′ ) to ℓ(Λ (n,m) , ∆).This maps the commuting quantum integrals in question to an algebra of commuting difference operators in ℓ(Λ (n,m) , ∆) containing H (2.2).Since [DEZ18,Theorem 9.5] guarantees that the latter algebra of commuting difference operators is (Harish-Chandra-)isomorphic to the algebra of complex functions on the joint spectrum {ξ κ | κ ∈ Λ (m,n) } ⊂ A m , this establishes the quantum integrability of our q-difference Toda hamiltonian H (2.2).