Linear Ordinary Differential Operators of the Second Order
Linear Ordinary Differential Operators of the Second Order
Ralph S Phillips
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L) (tlif. T is, in D), This incidentally, proves that R(X, L) satisfies Eq. (13). Finally we note that the conditions of the theorem are satisfied by the functions y such that y(0) = and y, y', and y*' are absolutely continuous and all vanish outside a finite inter- val. Hence these functions lie in D; and since they are dense in 1^(0, oo) then by our definition L is a well-defined operator, Theorez: 2. 2 . The resolvent E(X, L) exists for all Im(X) > and is given in the Tisual way by (11). It ...is clear that if the range of R(\, L) is contained in D then (2) will be satisfied. Let f e 1^(0, 00) and set y = R(X, L)f in Eq. (11). It is again clear that y, y* are absolutely continuous, that y and -y^' + q(x)y belong to 12(0, 00), and that y(0) = 0. Finally -9- V^ [y. H;(x, X^) ] = W^[v(x, \), v(x, \^) ] r ()(y. X) f(y) dy -'o + W^ |^()(x, X), v(x, \q) J [ vt/(y. X) f(y) dy. z Now in the limit circle case toth f (J(yiX) f (y) dy — ^ limit and 7 v(yiX) f (y) dy —4 0. We now make use of two results due to Titchmarsh [5] (d 22 V namely, (16) .
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