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We prove the orthogonality of the Bethe Ansatz eigenfunctions for theLaplacian on a hyperoctahedral Weyl alcove with repulsive homogeneous Robinboundary conditions at the walls. To this end these eigenfunctions areretrieved as the continuum limit of an orthogonal basis of algebraic BetheAnsatz eigenfunctions for a finite $q$-boson system endowed with diagonalopen-end boundary interactions.

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