Diagonalization of the infinite q-boson system

Abstract

Abstract. We present a hierarchy of commuting operators in Fock space containing the q-boson Hamiltonian on Z and show that the operators in question are simultaneously diagonalized by Hall-Littlewood functions. As an application, the n-particle scattering operator is computed. IntroductionThe q-boson model constitutes a one-dimensional exactly solvable particle system in Fock space [BIK] based on the q-oscillator algebra [KS, Ch. 5]. In the case of periodic boundary conditions (i.e. with particles hopping on the finite lattice Z m ), the integrability, the spectrum, and the eigenfunctions of the Hamiltonian were analyzed by means of the algebraic Bethe Ansatz method [BIK]. Remarkably, these eigenfunctions turn out to be Hall-Littlewood functions [T, K] (cf. also [J] for an alternative construction of Hall-Littlewood functions in Fock space based on deformed vertex operator algebras, with applications in the study of KP τ -functions arising from generating functions of weighted plane partitions [FW]). With the aid of explicit expressions for the commuting quantum integrals arising from an infinite-dimensional solution of the Yang-Baxter equation, it was very recently demonstrated [K] that the eigenvalue problem for the q-boson system on Z m is in fact equivalent to that of an integrable discretization [D] of the celebrated delta Bose gas on the circle [LL].The present work addresses the spectral problem and the integrability of the q-boson system on the infinite lattice Z. Specifically, we demonstrate that the eigenfunctions of this infinite q-boson system are again given by Hall-Littlewood functions and provide explicit formulas for a complete hierarchy of operators commuting with the Hamiltonian; these formulas are natural infinite-dimensional analogues of the above-mentioned expressions in [K] for the finite q-boson system on Z m . Finally, the n-particle scattering operator is computed as an application of Ruijsenaars' general scattering results in [R2]. The infinite q-boson systemGiven n ≥ 0 integral, let F (Λ n ) be the space of complex functions f : Λ n → C on the fundamental domain of the integral lattice Z n modulo the action of the permutation group S n , where Λ 0 := {0} and F (Λ 0 ) := C by convention. We will refer to the infinite direct sumbuilt of all finite linear combinations of (arbitrary) functions f n ∈ F (Λ n ), n = 0, 1, 2, . . . as the algebraic Fock space. For λ ∈ Λ n and l ∈ Z, let the multiplicity m l (λ) count the number of components λ j , 1 ≤ j ≤ n such that λ j = l. We write β * l λ for the point of Λ n+1 obtained from λ by inserting an additional component with value l and-assuming m l (λ) > 0-we write β l λ ∈ Λ n−1 for the result of the inverse operation that deletes a component with value l from λ. Upon defining the following actions on f ∈ F (Λ n ):3)where 0 < q < 1 andfor m = 0, 1, 2, . . ., it is readily verified that one ends up with a representation of the q-boson field algebra on F :for l = k, andHere the brackets refer to the (ordinary) commutator product. By constr...

Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz

“…9.4. q-Boson → Van Diejen's delta Bose gas. Another scaling [BCPS15, §6.2] of the q-Boson generator ( §9.1) takes us to a semi-discrete delta Bose gas studied by Van Diejen [vD04] (other models of a similar nature are discussed in [vDE14a], [vDE14b]). The limiting operator is equivalent to the following free operator subject to two-body boundary conditions (we use notation of §5.1)…”

Spectral theory for the -Boson particle system

“…Due to our probabilistic motivations, we primarily consider γ = −1 here, though in Section 6.1 we consider the general γ Hamiltonian (under the identification of γ = −ǫ) and show how our results extend. The spectral theory for the periodic γ = 0 case has received attention recently in [85,87,53], and likewise for the infinite lattice (as considered herein) γ = 0 case in [86,88,89]. In these cases, Hall-Littlewood polynomials play the role of left and right eigenfunctions.…”

Quantum Integrals for a Semi-Infinite q-Boson System with Boundary Interactions

“…For the q-boson systems on the finite periodic lattice and on the (bi-)infinite lattice, analogous descriptions of the commuting quantum integrals stemming from the Pieri formulas for the Hall-Littlewood functions can be found in [5,16,26] and in [7], respectively. Previously, Pieri formulas for Macdonald's (q-deformed Hall-Littlewood) polynomials were interpreted in a similar vein as eigenvalue equations for the quantum integrals of lattice Ruijsenaars-Schneider type models [4,10,22,23].…”

“…The q-boson system [1,24] constitutes an integrable q-deformed lattice regularization of the quantum nonlinear Schrödinger equation [11,15,17] built of q-oscillators [12,20]. Its n-particle Bethe Ansatz eigenfunctions amount to the celebrated Hall-Littlewood functions [7,16,26]. The model in question can moreover be viewed as a degeneration of the recently found stochastic q-Hahn particle system [2,21,25].…”