Anti-Locality of the One-half Power of Elliptic Differential Operators
Kyûya Masuda · Publications of the Research Institute for Mathematical Sciences · 1972
In connection with the question concerning the ring of operators generated by quantized field operators with a given region of space, using the edge of the wedge theorem, H. Reeh and S. Schlieder Q3] showed that the one-half power (m2 1— J)1/2 of m,2/— J(on RN) has the anti-local property in the sense that if / and (m2!- J)1/2/(/eL2CR^)) vanish in some nonempty open set U in RN, then /(#) must be identically zero in RN. I. Segal and R. Goodman [_4T generalized the result of Reeh and Schlieder, and showed that (m,2/— J)x (A: non-integral number) is also anti-local if the space dimension N is odd. Recently, M. Murata [_2~] succeeded, by the so-called method of decent, in excluding the assumption that the space dimension is odd. The purpose of the present paper is to give another proof for the theorem of Reeh and Schlieder on the antilocality of (m2!— J)1/2. The proof may be of some interest in two respects; it is simple, and secondly the method is applicable to the case of the one-half power of elliptic differential operators with variable coefficients, especially (V(x) — J)1/2 (V(x}: external potential). Now let Q be a domain with smooth boundary in an ^-dimensional Euclidean space RN. We define the operator A in L2(J2) by: ;u = Q on the boundary of Q}