Spectral theory of commuting self-adjoint partial differential operators |
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Institution: | Department of Mathematics, University of Iowa, Iowa City, Iowa 52242 U.S.A. |
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Abstract: | Let Ω denote a connected and open subset of n. The existence of n commuting self-adjoint operators H1,…, Hn on such that each Hj is an extension of (acting on is shown to be equivalent to the existence of a measure μ on n such that f → tf (the Fourier transform of f) is unitary from onto Ω. It is shown that the support of μ can be chosen as a subgroup of n iff H1,…, Hn can be chosen such that the unitary groups generated by H1,…, Hn act multiplicatively on . This happens iff Ω (after correction by a null set) forms a system of representatives for the quotient of n by some subgroup, i.e., iff Ω is essentially a fundamental domain. |
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