Orthogonality of Bethe Ansatz eigenfunctions for the Laplacian on a hyperoctahedral Weyl alcove

TL;DR

This paper proves the orthogonality of Bethe Ansatz eigenfunctions for the Laplacian in a hyperoctahedral Weyl alcove, connecting continuum eigenfunctions with finite q-boson system eigenfunctions through a continuum limit.

Contribution

It establishes the orthogonality of these eigenfunctions and links continuum and discrete models via a continuum limit process.

Findings

Eigenfunctions are orthogonal under Robin boundary conditions

Continuum eigenfunctions derived as limits of finite q-boson system eigenfunctions

Provides a rigorous mathematical foundation for Bethe Ansatz in this setting

Abstract

We prove the orthogonality of the Bethe Ansatz eigenfunctions for the Laplacian on a hyperoctahedral Weyl alcove with repulsive homogeneous Robin boundary conditions at the walls. To this end these eigenfunctions are retrieved as the continuum limit of an orthogonal basis of algebraic Bethe Ansatz eigenfunctions for a finite $q$-boson system endowed with diagonal open-end boundary interactions.