Boundary Interactions for the Semi-Infinite q-Boson System and Hyperoctahedral Hall-Littlewood Polynomials

We present a semi-infinite q-boson system endowed with a four-parameter boundary interaction. The n-particle Hamiltonian is diagonalized by generalized Hall-Littlewood polynomials with hyperoctahedral symmetry that arise as a degeneration of the Macdonald-Koornwinder polynomials and were recently studied in detail by Venkateswaran.


Remark 1.
To avoid possible confusion, it is important to emphasize that the parameter q of the q-boson model does not correspond to the q-deformation parameter that enters in Macdonald's theory of orthogonal polynomials associated with root systems [9,10] but rather to the parameter t used there. A different parameter t is employed below to abbreviate our notation for a frequently appearing product comprised by the four Askey-Wilson-type parameters t 1 , . . . , t 4 of the Macdonald-Koornwinder polynomial (and its (q → 0) Hall-Littlewood-type degeneration).

.1 Orthogonality
Let W be the hyperoctahedral group formed by the semi-direct product of the symmetric group S n and the n-fold product of the cyclic group Z 2 ∼ = {1, −1}. An element w = (σ, ) ∈ W acts naturally on ξ = (ξ 1 , . . . , ξ n ) ∈ R n via wξ := ( 1 ξ σ 1 , . . . , n ξ σn ) (with σ ∈ S n and j ∈ {1, −1} for j = 1, . . . , n). The algebra A of W -invariant polynomials on the torus T n := R n /(2πZ n ) is spanned by the hyperoctahedral monomial symmetric functions where Λ n stands for the set of partitions λ = (λ 1 , . . . , λ n ) ∈ Z n with the convention λ 1 ≥ · · · ≥ λ n ≥ 0, and the summation is meant over the orbit of λ with respect to the action of W ; the bracket ·, · refers to the standard inner product on R n , i.e. µ, ξ = µ 1 ξ 1 + · · · + µ n ξ n . The basis of hyperoctahedral Hall-Littlewood polynomials p λ (ξ), λ ∈ Λ n studied in [16] arises from the monomial basis via a (partial) Gram-Schmidt-like process as the trigonometric polynomials of the form such that Here we have employed the hyperoctahedral dominance partial ordering of the partitions (which differs from the usual dominance partial order in that one does not demand the additional degree homogeneity condition µ 1 + · · · + µ n = λ 1 + · · · + λ n for the partitions to be comparable) together with the following inner product on A: with |W | = 2 n n! denoting the order of the hyperoctahedral group and Throughout it is assumed that the parameters belong to the domain q ∈ (0, 1) and t r ∈ (−1, 1) \ {0}, r = 1, . . . , 4.
The hyperoctahedral Hall-Littlewood polynomials satisfy the following orthogonality relations [16]: where Here the multiplicity m l (λ) counts the number of parts λ j , 1 ≤ j ≤ n of size λ j = l (so m 0 (λ) is equal to n minus the number of nonzero parts) and we have used q-shifted factorials with the convention that (x) 0 = 1. Notice that the orthogonality p λ , p µ ∆ = 0 for distinct partitions λ and µ is manifest from the defining properties in equations (1) when both weights are comparable in the hyperoctahedral dominance partial ordering (2), whereas for noncomparable partitions the orthogonality is not at all obvious from the above construction.

Explicit formula
The orthogonality relations in equations (4) -which arise as a (q → 0) degeneration of wellknown orthogonality relations for the Macdonald-Koornwinder multivariate Askey-Wilson polynomials [3, 7, 10] -form a two-parameter extension of Macdonald's orthogonality relations for the Hall-Littlewood polynomials associated with the root system BC n [9, § 10]. An explicit formula for the hyperoctahedral Hall-Littlewood polynomials (1) generalizing the corresponding classic formula of Macdonald is given by [16] p λ (ξ) = 1 with and
Proposition 1 (Pieri formula). The normalized hyperoctahedral Hall-Littlewood polynomials P λ (ξ), λ ∈ Λ n satisfy the recurrence relation with the vectors e 1 , . . . , e n denoting the standard unit basis of Z n and Here we have employed the q-integers in the conditional sums on the r.h.s. of the recurrence stands for 'such that').
3 Boundary interactions for the semi-inf inite q-boson system 3.1 Deformed q-boson f ield algebra Let 2 (Λ n , N ) be the Hilbert space of functions f : Λ n → C determined by the inner product with N λ given by equation (4b) and the convention that Λ 0 := {∅} and 2 (Λ 0 , N ) := C. We think of 2 (Λ n , N ) as the Hilbert space for a system of n quantum particles on the nonnegative integer lattice N := {0, 1, 2, . . .} (i.e. the parts λ j , j = 1, . . . , n of λ ∈ Λ n encode the positions of the particles in question). In the Fock space consisting of all sequences n≥0 f n with f n ∈ 2 (Λ n , N ) such that n≥0 f n , f n n < ∞, we introduce bounded annihilation operators β l , l ∈ N that are perturbed at the boundary site = 0 and act on f ∈ 2 (Λ n , N ) via if n > 0, and β l f := 0 if n = 0. Here β * l λ ∈ Λ n is obtained from λ by adding a part of size l. The action on f ∈ 2 (Λ n , N ) of the adjoint of β l in H produces the creation operator if m l (λ) > 0, and (β * l f )(λ) = 0 otherwise. Here β l λ ∈ Λ n is obtained from λ with m l (λ) > 0 by discarding a part of size l. In the present setting, the role of the number operators is played by the bounded multiplication operators When t = q m for m = 1, 2, 3, . . ., the creation and annihilation operators β * l , β l together with the commuting operators N l , where l ∈ N and c ∈ Z) represent a four-parameter deformation of the q-boson field algebra at the boundary sites l = 0 and l = 1: and for l < k and Indeed, it is straightforward to verify the commutation relations in equations (11) upon computing the explicit actions of both sides on an arbitrary function f ∈ 2 (Λ n , N ) with the aid of the formulas in equations (9) and (10).

Hamiltonian
The Hamiltonian of our semi-infinite q-boson system with boundary interaction is of the form where V (N 0 , N 1 ) denotes a boundary potential that depends rationally on N 0 and N 1 . By construction, H (12) preserves the n-particle sector 2 (Λ n , N ) ⊂ H and we will denote the restriction of the Hamiltonian to this n-particle subspace by H n .

Diagonalization
From now on we will pick the boundary potential V (N 0 , N 1 ) in H (12) of the form By writing the action of V (N 0 , N 1 ) (14) on an arbitrary f ∈ 2 (Λ n , N ) as a rational expression in the parameters t r (r = 1, . . . , 4), it is readily seen -upon canceling possible common factors in the numerators and denominators -that V (N 0 , N 1 ) constitutes a bounded multiplication operator in 2 (Λ n , N ). It follows moreover from the Pieri recurrence in Proposition 1 and the explicit formula for H n in Proposition 2 that the Hamiltonian with this boundary potential is diagonalized in the n-particle subspace by a hyperoctahedral Hall-Littlewood wave function φ ξ : Λ n → C of the form where ξ ∈ T n plays the role of the spectral parameter.
Remark 2. The diagonalization in Proposition 3 in terms of the hyperoctahedral Hall-Littlewood polynomials implies that our q-boson Hamiltonian H n is unitarily equivalent to a multiplication operator governed by the eigenvalue E n (ξ) (16). A complete system of commuting quantum integrals for H n is obtained via this unitary equivalence from the multiplication operators associated with the elements of the algebra A of W -invariant trigonometric polynomials on T n . It remains an open problem to present an explicit construction in the spirit of [4] that lifts H (12) with V (N 0 , N 1 ) given by equation (14) to an infinite hierarchy of commuting operators in the Fock space H (8), reproducing the quantum integrals of H n upon restriction to the n-particle subspace 2 (Λ n , N ).

Ultralocality and coordinate Bethe ansatz
For general parameter values the deformation of the q-boson field algebra in Section 3.1 fails to be ultralocal, as the commutativity between the creation and annihilation operators at sites l = 0 and l = 1 is broken. The commutativity (and hence ultralocality) is restored when at least one of the four boundary parameters t r tends to zero (so t → 0). It is furthermore clear from the explicit expression in equations (5) for the hyperoctahedral Hall-Littlewood polynomial p λ (1) that the wave function φ ξ (15) fails to be of the usual coordinate Bethe ansatz form (at the boundary), as the expansion coefficients C λ (wξ) of the plane waves e −i wξ,λ depend on (the number of nonzero parts of) λ. By letting at least two of the four boundary parameters t r tend to zero the polynomial p λ (1) reduces to Macdonald's Hall-Littlewood polynomial associated with the root system of type BC, which implies that in this limiting case it is possible to rewrite the wave function in the conventional Bethe ansatz form. We end up by detailing our construction for these three-and two-parameter specializations of the boundary interaction.
modeling a system of n impenetrable bosons on N. In [5,Section 5] it was shown that the largetimes asymptotics of the q-boson dynamics generated by H n (19) is related to the impenetrable boson dynamics of H n,0 (3) via an n-particle scattering matrix of the form with The discussion in [5, Section 5] applies verbatim to our more general Hamiltonian H n from Proposition 2 with V (N 0 , N 1 ) given by equation (14), upon replacing s 0 (x) (20b) by This reveals that the n-particle scattering matrix of the model factorizes in two-particle bulk scattering matrices s(·) governed by a coupling parameter q and one-particle boundary scattering matrices s 0 (·) governed by coupling parameters t 1 , . . . , t 4 .