14.—An Example concerning the Separation Property of Differential Operators

Abstract

SynopsisThe differential expression M[f] = −f″+qf, on a half-line [a, ∞), is said to be ‘separated’ in L2(a, ∞) if the collection of all functions f ∈ L2(a, ∞) such that M[f] is defined and also in L2(a, ∞), has the property that both terms f″ and qf ar separately in L2(a, ∞). When q is positive and differentiable on [a, ∞) it is known that separation holds for M[·] if q satisfies the condition |q| ≦ on [a, ∞)(*) provided the constant c satisfies 0 < c < 2. This paper constructs a class of examples of the coefficient q to show that (*) does not necessarily yield separation if c > 4/√3>2. The precise upper bound of c for which (*) yields separation is not known.