Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Spectral asymptotics for Sturm-Liouville equations with indefinite weight

Authors: Paul A. Binding, Patrick J. Browne and Bruce A. Watson
Journal: Trans. Amer. Math. Soc. 354 (2002), 4043-4065
MSC (2000): Primary 34L20, 34B09, 34B24; Secondary 47E05
Published electronically: May 22, 2002
MathSciNet review: 1926864
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Sturm-Liouville equation

\begin{displaymath}-(py')' + qy =\lambda ry \text{\rm on} [0,l]\end{displaymath}

is considered subject to the boundary conditions

\begin{displaymath}y(0)\cos\alpha = (py')(0)\sin\alpha,\end{displaymath}

\begin{displaymath}y(l)\cos\beta = (py')(l)\sin\beta.\end{displaymath}

We assume that $p$ is positive and that $pr$ is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to $O(1/\sqrt{n})$for $\sqrt{\lambda_n}$, or equivalently up to $O(\sqrt{n})$ for $\lambda_n$, the eigenvalues of the above boundary value problem.

References [Enhancements On Off] (What's this?)

  • 1. F. V. Atkinson, A. B. Mingarelli, Asymptotics of the number of zeros and the eigenvalues of general weighted Sturm-Liouville problems, J. reine angewandte Math. (1987) 375/376, 380-393. MR 88d:34023
  • 2. P. A. Binding, P. J. Browne, K. Seddighi, Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proc. Edinburgh Math. Soc. (1993) 37, 57-72. MR 95k:34039
  • 3. E. A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955. MR 16:1022b
  • 4. A. A. Dorodnicyn, Asymptotic laws of distribution of the characteristic values for certain special forms of differential equations of the second order, Amer. Math. Soc. Translations, Series 2, (1960) 16, 1-101. MR 22:8161
  • 5. W. Eberhard, G. Freiling, The distribution of the eigenvalues for second order eigenvalue problems in the presence of an arbitrary number of turning points, Results in Math. (1992) 21, 24-41. MR 93e:34108
  • 6. G. M. Guabreev, V. N. Pivovarchik, Spectral analysis of the Regge problem with parameters, Funct. Anal. Appl. (1997) 31, 54-57. MR 98g:34134
  • 7. H. Hochstadt, On inverse problems associated with Sturm-Liouville operators, J. Differential Equations (1975) 17, 220-235. MR 51:3601
  • 8. E. L. Ince, Ordinary Differential Equations, Dover, 1944. MR 6:65f
  • 9. R. E. Langer, On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order, Trans. Amer. Math. Soc. (1931) 33, 23-64.
  • 10. B. M. Levitan, M. G. Gasymov, Determination of a differential equation by two of its spectra, Russian Math. Surveys (1964) 19, no. 2, 1-64. MR 29:299
  • 11. S. Strelitz, Asymptotics for solutions of linear differential equations having turning points with applications, Mem. Amer. Math. Soc. (1999) 142, Number 676. MR 2000e:34086

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 34L20, 34B09, 34B24, 47E05

Retrieve articles in all journals with MSC (2000): 34L20, 34B09, 34B24, 47E05

Additional Information

Paul A. Binding
Affiliation: Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada T2N 1N4

Patrick J. Browne
Affiliation: Mathematical Sciences Group, Department of Computer Science, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 5E6

Bruce A. Watson
Affiliation: Department of Mathematics, University of the Witwatersrand, Private Bag 3, P O WITS 2050, South Africa

Keywords: Eigenvalue asymptotics, indefinite weight, turning point
Received by editor(s): January 12, 2002
Published electronically: May 22, 2002
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society