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

  • W. N. Everitt (a1) and M. Giertz (a2)

Synopsis

The 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 fL2(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.

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[1]Atkinson, F. V., 1973. On some results of Everitt and Giertz. Proc. Roy. Soc. Edinb., 71A, 151158.
[2]Everitt, W. N. and Giertz, M., 1971. Some properties of the domains of certain differential operators. Proc, Lond. Math. Soc., 3, 301324.
[3]Everitt, W. N. 1972. Some inequalities associated with certain ordinary differential operators. Math. Z., 126, 308326.
[4]Everitt, W. N., Inequalities and separation criteria for certain differential operators (to be published).
[5]Everitt, W. N., Giertz, M. and Weidmann, J. Some remarks on a separation and limit-point criterion of second-order ordinary differential expressions. Math. Annln (in press).
[6]Naimark, M. A., 1968. Linear Differential Operators: Part II. New York: Ungar.
[7]Titchmarsh, E. C., 1962. Eigenfunction expansions associated with second-order differential equations: Part I (2nd edn). O.U.P.

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

  • W. N. Everitt (a1) and M. Giertz (a2)

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