22.—A Critical Class of Examples concerning the Integrable-square Classification of Ordinary Differential Equations

Abstract

SynopsisLet the coefficient q be real-valued on the half-line [0, ∞) and let q′ be locally absolutely continuous on [0, ∞). The ordinary symmetric differential expressions M and M2 are determined byIt has been shown in a previous paper by the authors that if for non-negative numbers k and X the coefficient q satisfies the conditionthen M is limit-point and M2 is limit–2 at ∞.This paper is concerned with showing that for powers of the independent variable x the condition (*) is best possible in order that both M and M2 should have the classification at ∞ given above.