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Spectral theory of finite volume domains in Rn
Journal article   Open access   Peer reviewed

Spectral theory of finite volume domains in Rn

Palle E.T Jørgensen
Advances in mathematics (New York. 1965), Vol.44(2), pp.105-120
1982
DOI: 10.1016/0001-8708(82)90001-9
url
https://doi.org/10.1016/0001-8708(82)90001-9View
Published (Version of record) Open Access

Abstract

Let Ω be an arbitrary open subset of R n of finite positive measure, and assume the existence of a subset Λ ⊂ R n such that the exponential functions e λ = exp i ( λ 1 x 1 + … + λ n x n ), λ = ( λ 1 ,…, λ n ) ∈ Λ , form an orthonormal basis for L 2 (Ω) with normalized measure. Assume 0 ∈ Λ and define subgroups K and A of ( R n , +) by K = Λ 0 = {γ ∈ R n :γ·λ ∈ 2π Z }, A = {a ∈ R n :U a m U∗ a = m }, where U t is the unitary representation of R n on L 2 (Ω) given by U t e = e itλ e λ , t ∈ R n , λ ∈ Λ, and where m is the multiplication algebra of L ∞ (Ω) on L 2 . Assume that A is discrete. Then there is a discrete subgroup D ⊃ A of dimension n, a fundamental domain D for D, and finite sets of representers R Λ , R Γ , R Ω , each containing 0, R Λ for A K in K 0 , and R Ω for A K in A such that Ω is disjoint union of translates of D : Ω = ∪ a∈ R Ω ( a + D ), neglecting null sets, and Λ = R Λ ⊕ D 0 . If R Γ is a set of representers for D A in D, then Γ = R Γ ⊕ K is a translation set for Ω, i.e., Ω ⊕ Γ = R n , direct sum, (neglecting null sets). The case A = R n corresponds to Ω = D , Λ = D 0 and Γ = K . This last case corresponds in turn to a function theoretic assumption of Forelli.

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