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
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.
KeywordsSturm-Liouville equation boundary condition Prüfer transformation.
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.
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