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The asymptotic form of the Titchmarsh–Weyl coefficient for a fourth order differential equation
Published online by Cambridge University Press: 14 November 2011
Synopsis
This paper considers the asymptotic form, as λ tends to infinity in sectors omitting the real axis, of the matrix Titchmarsh-Weyl coefficient M(λ) for the fourth order equation y(4) + q(x)y = λy, where q(x) is real and locally absolutely integrable. By letting M0(λ) denote the m-coefficient for the Fourier case y(4) = λy, the asymptotic formula M(λ) = M0(λ) + 0(1) is established.
- Type
- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 97 , 1984 , pp. 109 - 123
- Copyright
- Copyright © Royal Society of Edinburgh 1984
References
1Atkinson, F. V.. On the location of the Weyl circles. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 345–356.CrossRefGoogle Scholar
2Barrett, J. H.. Oscillation theory of ordinary linear differential equations. Adv. in Math. 3 (1969), 415–509.CrossRefGoogle Scholar
3Chaudhuri, J. and Everitt, W. N.. On the spectrum of ordinary second order differential operators. Proc. Roy. Soc. Edinburgh Sect. A 68 (1968), 95–119.Google Scholar
4Everitt, W. N.. Fourth order singular differential equations. Math. Ann. 149 (1963), 320–340.CrossRefGoogle Scholar
5Everitt, W. N.. On a property of the m-coefficient of a second order linear differential equation. J. London Math. Soc. 4 (1972), 443–457.CrossRefGoogle Scholar
6Everitt, W. N. and Halvorsen, S. G.. On the asymptotic form of the Titchmarsh-Weyl m-coefficieht. Applicable Anal. 8 (1978), 153–169.CrossRefGoogle Scholar
7Everitt, W. N. and Bennewitz, C.. Some remarks on the Titchmarsh–Weyl m-coefficient. In Tribute to Åke Pleijel, pp. 99–108. (Mathematics Department, University of Uppsala, Sweden, 1980).Google Scholar
8Harris, B. J.. On the Titchmarsh–Weyl m-function. Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 223–238.CrossRefGoogle Scholar
9Hille, E.. Lectures on Ordinary Differential Equations (Reading, Mass.: Addison-Wesley, 1969).Google Scholar
10Hinton, D. B. and Shaw, J. K.. On Titchmarsh–Weyl m-functions for Hamiltonian systems. J. Differential Equations 40 (1981), 316–342.CrossRefGoogle Scholar
11Hinton, D. B. and Shaw, J. K.. On boundary value problems for Hamiltonian systems with two singular points. SIAM J. Math. Anal. to appear.Google Scholar
12Hinton, D. B. and Shaw, J. K.. On the spectrum of a singular Hamiltonian system. Quaestiones Math. 5 (1982), 29–81.CrossRefGoogle Scholar
13Hinton, D. B. and Shaw, J. K.. Titchmarsh–Weyl theory for Hamiltonian systems. In Spectral Theory of Differential Operators, pp. 219–231 (Knowles, I. W. and Lewis, R. T. editors) (New York: North-Holland, 1981).Google Scholar
14Pinto, J. S. and Wood, A. D.. On the asymptotic behavior of the Titchmarsh–Weyl coefficients for a fourth order equation. Quaestiones Math. 4 (1981), 337–345.CrossRefGoogle Scholar
15Titchmarsh, E. C.. Eigenfunction Expansions Associated with Second order Differential Equations, Part 1, 2nd edn (Oxford: Clarendon Press, 1962).CrossRefGoogle Scholar