Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-20T17:25:38.604Z Has data issue: false hasContentIssue false
  • English
  • Français

13.—On Some Results of Everitt and Giertz

Published online by Cambridge University Press:  14 February 2012

F. V. Atkinson
F. V. Atkinson
Affiliation:
University of Toronto.
Get access Rights & Permissions [Opens in a new window]

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 f ∈ L2(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 q′q−3/2 on [a, ∞).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 71 , Issue 2 , 1973 , pp. 151 - 158
Copyright
Copyright © Royal Society of Edinburgh 1973

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References to Literature

[1]Atkinson, F. V., 1957. Asymptotic formulae for linear oscillations. Proc. Glasg. Math. Ass., 3, 105–111.CrossRefGoogle Scholar
[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, 199–204.Google Scholar
[3]Coppel, W. A., 1965. Stability and Asymptotic Behaviour of Differential Equations. Boston: Heath.Google Scholar
[4]Everitt, W. N. and Giertz, M., 1971. Some properties of the domains of certain differential operators. Proc. Lond. Math. Soc., 23, 301–324.Google Scholar
[5]Everitt, W. N. 1972. Some inequalities associated with certain ordinary differential operators. Math. Z., 126, 308–326.CrossRefGoogle Scholar
[6]Everitt, W. N. 1972. On some properties of the powers of a formally self-adjoint differential expression. Proc. Lond. Math. Soc., 24, 149–170.CrossRefGoogle Scholar
[7]Everitt, W. N. 1972. On some properties of the domains of powers of certain differential operators. Proc. Lond. Math. Soc., 24, 756–768.CrossRefGoogle Scholar
[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, 53–73.CrossRefGoogle Scholar
[9]Everitt, W. N. and Giertz, M., 1973. An example concerning the separation property for differential operators. Proc. Roy. Soc. Edinb., 71A, 159–165.Google Scholar