# Asymptotic formulae for eigenvalues of limit circle problems on a half line

- 58 Downloads
- 2 Citations

## Summary

For the classical limit-circle eigenvalue problem for −y″+qy=λy on [α, ∞) asymptotic formulae for the eigenvalues are obtained. For the positive spectrum a Prüfer transformation and an iterative procedure of Atkinson are employed to obtain higher order terms in the asymptotic expansions of λ_{n}. For the negative spectrum a piecewise turningpoint analysis making use of a linear approximation to q in the neighborhood of the turning point and a WKB approximation away from the turning point is employed to obtain first approximations to both the eigenvalues and eigenfunctions. In both cases assumptions are placed on q, and the turning-point analysis gives rise to slightly stronger assumptions in the case of the negative spectrum. An example of Titchmarsh, q=−exp (2x (for which solutions are available for all values of the eigenvalue parameter in terms of Bessel functions), together with the limit-circle theory of Fulton, provides an independent verification of the general results for a specific case; in particular, Olver's uniform asymptotic expansion for J_{v}(vz) as v→∞ is used to double-check the asymptotic formula for the eigenfunctions in the case of the negative spectrum.

### Keywords

Asymptotic Expansion Eigenvalue Problem Bessel Function Linear Approximation Iterative Procedure## Preview

Unable to display preview. Download preview PDF.

### References

- [1]M.Abramowitz- I. A.Stegun,
*Handbook of mathematical functions*, NBS Applied Math. Series 55, U. S. Department of Commerce, 1964.Google Scholar - [2]A. G. Alenitsyn,
*Asymptotic properties of the spectrum of a Sturm-Liouville operator in the case of a limit circle*, Differential Equations,**12**, no. 2 (1977), pp. 298–305. (Differentsial'nye Uravneniya,**12**, no. 3 (1976), pp. 428–437).Google Scholar - [3]F. V. Atkinson,
*On second-order linear oscillators*, Revista Tucuman,**8**(1951), pp. 71–87.Google Scholar - [4]F. V. Atkinson,
*Asymptotic formulae for linear oscillations*, Proc. Glasgow Math. Association,**3**(1957), pp. 105–111.Google Scholar - [5]V. P. Belogrud -A. G. Kostyuchenko, Usp. Mat. Nauk.,
**28**, no. 2 (170) (1973), pp. 227–228.Google Scholar - [6]W. A. Coppel,
*Stability and asymptotic behaviour of differential equations*, Heath and Co., Boston, 1965.Google Scholar - [7]
- [8]M. A. Evgrafov,
*Asymptotic estimates and entire functions*, Gordon and Breach, New York, 1961.Google Scholar - [9]G. Fix,
*Asymptotic eigenvalues of Sturm-Liouville systems*, J. Math. Anal. and Appls.,**19**(1967), pp. 519–525.Google Scholar - [10]C. Fulton,
*Parametrizations of Titchmarsh's m(λ)-functions in the limit circle case*, Trans. A.M.S.,**229**(1977), pp. 51–63.Google Scholar - [11]M. Giertz,
*On the solutions in L*_{2}(−∞, ∞)*of y″*+ (λ−q*(x*)y=0,*when q is rapidly increasing*, Proc. London Math. Soc., (3)**14**(1964), pp. 53–73.Google Scholar - [12]P. Hartman,
*Ordinary Differential Equations*, Wiley, Baltimore, 1973. (Corrected Reprint of original 1964 Edition).Google Scholar - [13]P. Heywood,
*On the asymptotic distribution of eigenvalues*, Proc. London Math. Soc., (3)**4**(1954), pp. 456–470.Google Scholar - [14]H. Hochstadt,
*Asymptotic estimates for the Sturm-Liouville spectrum*, Comm. Pure and Appl. Math.,**14**(1961), pp. 749–764.Google Scholar - [15]J. Horn,
*über lineare Differentialgleichungen mit einem verÄnderlichen Parameter*, Math. Ann.,**52**(1899), pp. 340–362.Google Scholar - [16]W. Magnus -F. Oberhettinger -R. P. Soni,
*Formulas and theorems for the special functions of mathematical physics*, Third Edition, Springer-Verlag, New York, 1966.Google Scholar - [17]F. W. J. Olver,
*Error bounds for first approximations in turning point problems*, SIAM J. Appl. Math.,**11**(1963), pp. 748–772.Google Scholar - [18]F. W. J. Olver,
*Error bounds for asymptotic expansions in turning point problems*, SIAM J. Appl. Math.,**12**(1964), pp. 200–214.Google Scholar - [19]F. W. J. Olver,
*The asymptotic expansion of Bessel functions of large order*, Phil. Trans., A**247**(1954), pp. 328–368.Google Scholar - [20]
- [21]E. C. Titchmarsh,
*Eigenfunction expansions associated with second-order differential equations*, part I, Second Edition, Clarendon Press, Oxford, 1962.Google Scholar - [22]G. N. Watson,
*Theory of Bessel Functions*, Second Edition, Cambridge University Press, London, 1944.Google Scholar