A generalization to locally compact abelian groups of a spectral problem for commuting partial differential operators

Abstract

Let G be a locally compact abelian group, and let Ω be an open relatively compact subset of positive Haar measure. Let Λ be a subset of the dual group G such that the restriction to Ω of Λ(·) for λϵΛ constitute an orthonormal basis for L 2(Ω) with normalized measure. We show that the pair Ω,Λ can be characterized completely in terms of group theory and the geometry of fundamental domains for discrete subgroups. Proofs are only sketched. The relationship to partial differential operators is pointed out.