Semi-boundedness of Ordinary Differential Operators

The boundedness from above and from below of very general symmetric quasi-differential operators S is studied. Here we show that (i) if S is regular, of even order, and has positive leading coefficient, then it is bounded below; (ii) if the order of S (which may be regular or singular) is odd, then S is unbounded both above and below; and (iii) if S is of even order, regular or singular, but with a leading coefficient which changes sign, then S is unbounded both above and below.