A note on a paper of Harris concerning the asymptotic approximation to the eigenvalues of -y'' + qy = λy, with boundary conditions of general form
© Hormozi; licensee Springer. 2012
Received: 19 October 2011
Accepted: 12 April 2012
Published: 12 April 2012
The Erratum to this article has been published in Boundary Value Problems 2017 2017:100
In this article, we derive an asymptotic approximation to the eigenvalues of the linear differential equation
with boundary conditions of general form, when q is a measurable function which has a singularity in (a, b) and which is integrable on subsets of (a, b) which exclude the singularity.
Mathematics Subject Classification 2000: Primary, 41A05; 34B05; Secondary, 94A20.
Our aim here is to obtain a formula like (1.7) in which the O(1) term is replaced by an integral term plus and error term of smaller order. We obtain an error term of . To achieve this we first use the differential Equation (1.6) to obtain estimates for θ(b) - θ(a) for general λ as λ → ∞.
2. Statement of result
as λ → ∞.
By using Lemmas 5.1 and 5.2 of  we conclude the following lemma
Now, we prove an elementary lemma.
Lemma 2.3. If g ∈ Ł1andthen
Remark 2.4. Lemma 2.2 shows that ifthen
This ends the proof of Lemma 2.5.
as λ → ∞.
When N = 1, applying Lemma 2.5, . Now By using Lemma 2.3 and induction we achieve that as λ → ∞.
Remark 2.7. By using the discussions of choice of f in, the condition (2.3) let us to consider q as the form q(x) ~ x-Kwhere 1 ≤ K < 2.
The author would like to thank Professor Grigori Rozenblum for useful comments.
- Atkinson FV, Fulton CT: Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity I. Proc R Soc Edinburgh Sect A 1984, 99(1-2):51-70. 10.1017/S0308210500025968MathSciNetView ArticleGoogle Scholar
- Atkinson FV: Asymptotics of an eigenvalue problem involving an interior singularity. In Research program proceedings ANL-87-26, 2. Argonne National Lab. Illinois; 1988:1-18.Google Scholar
- Harris BJ: Asymptotics of eigenvalues for regular Sturm-Liouville problems. J Math Anal Appl 1994, 183: 25-36. 10.1006/jmaa.1994.1128MathSciNetView ArticleGoogle Scholar
- Harris BJ: A note on a paper of Atkinson concerning the asymptotics of an eigenvalue problem with interior singularity. Proc Roy Soc Edinburgh Sect A 1988, 110(1-2):63-71. 10.1017/S0308210500024859MathSciNetView ArticleGoogle Scholar
- Harris BJ, Race D: Asymptotics of eigenvalues for Sturm-Liouville problems with an interior singularity. J Diff Equ 1995, 116(1):88-118. 10.1006/jdeq.1995.1030MathSciNetView ArticleGoogle Scholar
- Harris BJ, Marzano F: Eigenvalue approximations for linear periodic differential equations with a singularity. Electron J Qual Theory Diff Equ 1999, 1-18. No. 7Google Scholar
- Coskun H, Bayram N: Asymptotics of eigenvalues for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition. J Math Anal Appl 2005, 306(2):548-566. 10.1016/j.jmaa.2004.10.030MathSciNetView ArticleGoogle Scholar
- Fix G: Asymptotic eigenvalues of Sturm-Liouville systems. J Math Anal Appl 1967, 19: 519-525. 10.1016/0022-247X(67)90009-1MathSciNetView ArticleGoogle Scholar
- Fulton CT, Pruess SA: Eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems. J Math Anal Appl 1994, 188(1):297-340. 10.1006/jmaa.1994.1429MathSciNetView ArticleGoogle Scholar
- Fulton CT: Two point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc Roy Soc Edinburgh Sect A 1977, 77: 293-308.MathSciNetView ArticleGoogle Scholar
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