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It is only for a special subset of the natural boundary conditions for the operator d4w Aw=df that its positive square root is the negative second derivative operator. In this paper we develop a procedure for parametric description of all natural boundary conditions, we show which ones admit A1/2 in the form just noted, and we show that in the other cases -D2w = -4S = [/ + P]Ai/2w dxl where P is a bounded, but in general not compact, operator on the Hilbert space L2 [0,7i]. Possibledoi:10.1090/qam/973388 fatcat:jiapcifirbdjfiaiysgmocbpxu