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13.—On Some Results of Everitt and Giertz

  • F. V. Atkinson (a1)

Synopsis

The differential expression Mf = −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 Mf is defined and also in L2(a, ∞), has the property that both the terms f″ and qf are separately in L2(a, ∞). When q is positive and differentiable on [a, ∞) this paper obtains sufficient conditions on the coefficient q for M to be separated; these take the form of bounds for qq−3/2 on [a, ∞).

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[1]Atkinson, F. V., 1957. Asymptotic formulae for linear oscillations. Proc. Glasg. Math. Ass., 3, 105111.
[2]Chisholm, R. S. and Everitt, W. N., 1971. On bounded integral operators in the space of integrable-square functions. Proc. Roy. Soc. Edinb., 69A, 199204.
[3]Coppel, W. A., 1965. Stability and Asymptotic Behaviour of Differential Equations. Boston: Heath.
[4]Everitt, W. N. and Giertz, M., 1971. Some properties of the domains of certain differential operators. Proc. Lond. Math. Soc., 23, 301324.
[5]Everitt, W. N. 1972. Some inequalities associated with certain ordinary differential operators. Math. Z., 126, 308326.
[6]Everitt, W. N. 1972. On some properties of the powers of a formally self-adjoint differential expression. Proc. Lond. Math. Soc., 24, 149170.
[7]Everitt, W. N. 1972. On some properties of the domains of powers of certain differential operators. Proc. Lond. Math. Soc., 24, 756768.
[8]Giertz, M., 1964. On the solutions in L 2(−∞, ∞) of y″ + (λ−q(x))y = 0 when q is rapidly increasing. Proc Lond. Math. Soc., 14, 5373.
[9]Everitt, W. N. and Giertz, M., 1973. An example concerning the separation property for differential operators. Proc. Roy. Soc. Edinb., 71A, 159165.

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