The Expansion Theorem For Singular Self-Adjoint Linear Differential Operators

Abstract

1. There has recently appeared three papers giving independently and almost simultaneously a much simplified approach to the expansion theorem for selfadjoint singular ordinary differential equations. Levitan [7] treats by means of a simple lemma the differential operator of arbitrary order but does not deal with the problem of uniqueness of the expansion. Yosida [11] and Levinson [5, 6] considered only the second order case but with uniqueness proved in the limit point case. All three accounts make important use of the Helly selection theorem for a set of functions of bounded variation. Here it will be shown that using the lemma of Levitan [7] and a treatment rather similar to that used by Levinson [6] for the inverse transform theorem, it is possible to obtain a simple self-contained proof of uniqueness under broad conditions. In particular none of the apparatus of Hilbert space or the theory of singular integral equations will be required. One of the results that follow, Theorem II, has been demonstrated by Coddington [2] with the use of Levitan's lemma and theorems from Hilbert space theory. Moreover Theorem III is implicit in Coddington's results because of the remarks which follow his Theorem 3 [2, p. 735]. The methods of the present paper can easily be carried over to singular selfadjoint systems using the formulations given by Bliss [1] for the non-singular