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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Sturm-Liouville equation

is considered subject to the boundary conditions

We assume that is positive and that is piecewise continuous and changes sign at its discontinuities. We give asymptotic approximations up to for , or equivalently up to for , the eigenvalues of the above boundary value problem.

**1.**F. V. Atkinson and A. B. Mingarelli,*Asymptotics of the number of zeros and of the eigenvalues of general weighted Sturm-Liouville problems*, J. Reine Angew. Math.**375/376**(1987), 380–393. MR**882305****2.**P. A. Binding, P. J. Browne, and K. Seddighi,*Sturm-Liouville problems with eigenparameter dependent boundary conditions*, Proc. Edinburgh Math. Soc. (2)**37**(1994), no. 1, 57–72. MR**1258031**, 10.1017/S0013091500018691**3.**Earl A. Coddington and Norman Levinson,*Theory of ordinary differential equations*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR**0069338****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. Transl. (2)**16**(1960), 1–101. MR**0117381****5.**W. Eberhard and G. Freiling,*The distribution of the eigenvalues for second order eigenvalue problems in the presence of an arbitrary number of turning points*, Results Math.**21**(1992), no. 1-2, 24–41. MR**1146634**, 10.1007/BF03323070**6.**G. M. Gubreev and V. N. Pivovarchik,*Spectral analysis of the Regge problem with parameters*, Funktsional. Anal. i Prilozhen.**31**(1997), no. 1, 70–74 (Russian); English transl., Funct. Anal. Appl.**31**(1997), no. 1, 54–57. MR**1459834**, 10.1007/BF02466004**7.**Harry Hochstadt,*On inverse problems associated with Sturm-Liouville operators*, J. Differential Equations**17**(1975), 220–235. MR**0367359****8.**E. L. Ince,*Ordinary Differential Equations*, Dover Publications, New York, 1944. MR**0010757****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 and M. G. Gasymov,*Determination of a differential equation by two spectra*, Uspehi Mat. Nauk**19**(1964), no. 2 (116), 3–63 (Russian). MR**0162996****11.**S. Strelitz,*Asymptotics for solutions of linear differential equations having turning points with applications*, Mem. Amer. Math. Soc.**142**(1999), no. 676, viii+89. MR**1625068**, 10.1090/memo/0676

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

Email:
binding@ucalgary.ca

**Patrick J. Browne**

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

Email:
browne@snoopy.usask.ca

**Bruce A. Watson**

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

Email:
watson-ba@e-math.ams.org

DOI:
http://dx.doi.org/10.1090/S0002-9947-02-03023-4

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