Bethe Ansatz for the Ruijsenaars Model of BC₁-Type

Abstract

Abstract. We consider one-dimensional elliptic Ruijsenaars model of type BC 1 . It is given by a three-term difference Schrödinger operator L containing 8 coupling constants. We show that when all coupling constants are integers, L has meromorphic eigenfunctions expressed by a variant of Bethe ansatz. This result generalizes the Bethe ansatz formulas known in the A 1 -case.

Hilbert-Schmidt Operators vs. Integrable Systems of Elliptic Calogero-Moser Type IV. The Relativistic Heun (van Diejen) Case

“…The integral now continues analytically to Im x − σ ∈ (0, a s ). Since a s ≥ m and ρ (1) n (γ ; x) is holomorphic for Im x − σ ∈ (−m, m) (as already shown), the lemma follows.…”

“…In the relativistic elliptic BC 1 case at hand, there are only eigenfunction formulae available for an R 8 -lattice of coupling constants. Indeed, for this discrete set of couplings eigenfunctions of Floquet-Bloch type for one of the A∆Os were found by Chalykh [1]. They involve a transcendental system of 'Bethe Ansatz' equations whose solutions are quite inaccessible.…”

“…, γ 7 over multiples of iπ/r. From this description it is already clear that the two operators A ± (γ; x) leave the vector space of meromorphic and even functions invariant; furthermore, it is readily verified that for two arbitrary coupling vectors γ (1) , γ (2) , we need only require equality of component sums to obtain their commutativity:…”

“…Since (1.27) yields a huge family of commuting A∆O pairs, it is clearly too optimistic (and indeed quite wrong) to expect that there exist self-adjoint Hilbert space versions + (γ (1) ) and − (γ (2) ) that commute without restricting γ (1) and γ (2) . As we shall show, there do exist commuting self-adjoint Hilbert space versions when the couplings γ (1) and γ (2) are equal (and satisfy further restrictions), but there is no a priori guarantee that even this is feasible. We associate commuting self-adjoint operators ± (γ) on H to the A∆Os (1.23) in several stages.…”

Spectra of elliptic potentials and supersymmetric gauge theories

“…The example studied here is a non-relativistic two particles quantum mechanical problem with elliptic potential, has a direct connection with supersymmetric gauge theory. It has a relativistic generalization, the corresponding spectral problem is a difference equation with coefficients given by elliptic function [51][52][53]. Various relativistic or non-relativistic version of two particles quantum mechanical system with elliptic, trigonometric/hyperbolic and rational potentials are obtained from limits of the relativistic-DTV potential.…”