Spectral theory of commuting self-adjoint partial differential operators

Abstract

Let Ω denote a connected and open subset of R n . The existence of n commuting self-adjoint operators H 1,…, H n on L 2(Ω) such that each H j is an extension of − i∂ ∂x j (acting on C c ∞(Ω)) is shown to be equivalent to the existence of a measure μ on R n such that f → \\ ̂ tf (the Fourier transform of f) is unitary from L 2(Ω) onto Ω. It is shown that the support of μ can be chosen as a subgroup of R n iff H 1,…, H n can be chosen such that the unitary groups generated by H 1,…, H n act multiplicatively on L 2(Ω) . This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of R n by some subgroup, i.e., iff Ω is essentially a fundamental domain.