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Anti-locality of certain functions of the Laplace operator

In connection with non relativistic approximation of the relativistic quantum theory, I. Segal and R. Goodman [4] showed that the fractional power (m2I-4)2 (2; non-integral number) of (m2I-4) is anti-local in L2(En) when the space dimension n is odd, where the anti-locality means that if f and (m2I-4)2f (f E L2(En)) vanish in some non-empty open set U in En, then f (x) must be identically zero in En. The anti-locality of the operator (m2I-4)12 is also relevant to the quantum field theory, and the result of H. Reeh and S. Schlieder [2] is essentially equivalent to the anti-locality of the operator (m2I-4)112 (on En). Recently K. Masuda [1] generalized the result of H. Reeh and S. Schlieder, and showed