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A note on a paper of Harris concerning the asymptotic approximation to the eigenvalues of -y'' + qy = λy, with boundary conditions of general form

Mahdi Hormozi

Author affiliations

Department of Mathematical Sciences, Division of Mathematics, Chalmers University of Technology and University of Gothenburg, Gothenburg 41296, Sweden

Citation and License

Boundary Value Problems 2012, 2012:40  doi:10.1186/1687-2770-2012-40

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/40


Received:19 October 2011
Accepted:12 April 2012
Published:12 April 2012

© 2012 Hormozi; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this article, we derive an asymptotic approximation to the eigenvalues of the linear differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M1">View MathML</a>

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.

Keywords:
Sturm-Liouville equation; boundary condition; Prüfer transformation.

1. Introduction

Consider the linear differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M2">View MathML</a>

(1.1)

where λ is a real parameter and q is real-valued function which has a singularity in (a, b). According to [1], an eigenvalue problem may be associate with (1.1) by imposing the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M3">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M4">View MathML</a>

(1.3)

In [2], Atkinson obtained an asymptotic approximation of eigenvalues where y satisfies Dirichlet and Neumann boundary conditions in (1.1). Here, we find asymptotic approximation of eigenvalues for all boundary condition of the forms (1.2) and (1.3). To achieve this, we transform (1.1) to a differential equation all of whose coefficients belong to L1[a, b]. Then we employ a Prüfer transformation to obtain an approximation of the eigenvalues. In this way, many basic properties of singular problems can be inferred from the corresponding regular ones. In [3], Harris derived an asymptotic approximation to the eigenvalues of the differential Equation (1.1), defined on the interval [a, b], with boundary conditions of general form. But, he demands the condition, q Ll[a, b]. Atkinson and Harris found asymptotic formulae for the eigenvalues of spectral problems associated with linear differential equations of the form (1.1), where q(x) has a singularity of the form αx-k with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M6">View MathML</a> in [2,4] respectively. Harris and Race [5] generalized those results for the case 1 ≤ k < 2. In [6], Harris and Marzano derived asymptotic estimates for the eigenvalues of (1.1) on [0, a] with periodic and semi-periodic boundary conditions. The reader can find the related results in [7-10]. We consider q(x) = Cx-K where 1 ≤ K < 2 and an asymptotic approximation to the eigenvalues of (1.1) with boundary conditions of general form. Our technique in this article follows closely the technique used in [2-5]. Let U = [a, 0) ∪ (0, b] and q L1,Loc(U). As Harris did in [[5], p. 90], suppose that there exists some real function f on [a, 0) ∪ (0, b] in ACLoc([a, 0) ∪ (0, b]) which regularizes (1.1) in the following sense. For f which can be chosen in Section 2, define quasi-derivatives, y[i] as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M7">View MathML</a>

y is a solution of (1.1) with boundary conditions (1.2) and (1.3) if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M8">View MathML</a>

(1.4)

The object of the regularization process is to chose f in such way that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M9">View MathML</a>

(1.5)

Having rewritten (1.1) as the system (1.4), we observe that, for any solution y of (1.1) with λ > 0, according to [2,4], we can define a function θ AC(a, b) by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M10">View MathML</a>

When y[1] = 0, θ is defined by continuity [[5], p. 91]. It makes sense to mention that one can find full discussions and nice examples about the choice of f in [2,4,5]. Atkinson in [2] noticed that the function θ satisfies the differential equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M11">View MathML</a>

(1.6)

Let λ > 0 and the n-th eigenvalue λn of (1.1-1.3), then according to [[1], Theorem 2], Dirichlet and non-Dirichlet boundary conditions can be described as bellow:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M12">View MathML</a>

It follows from (1.5-1.6) that large positive eigenvalues of either the Dirichlet or non-Dirichlet problems over [a, b] satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M13">View MathML</a>

(1.7)

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M14">View MathML</a>. 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

We define a sequence ξj(t) for j = 1, ..., N + 1, t ∈ [a, b] by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M15">View MathML</a>

(2.1)

and note that in view of f, F L(a, b),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M16">View MathML</a>

(2.2)

Suppose that for some N ≥ 1,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M17">View MathML</a>

(2.3)

We define a sequence of approximating functions a

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M18">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M19">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M20">View MathML</a>

(2.6)

for j = 0, 1, 2, ... and for a x b. We measure the closeness of the approximation in the next result. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M21">View MathML</a>

(2.7)

The following lemma appears in [2,5].

Lemma 2.1. If g Ł1 then for any j and a x b

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M22">View MathML</a>

as λ → ∞.

By using Lemmas 5.1 and 5.2 of [5] we conclude the following lemma

Lemma 2.2. There exists a suitable constant C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M23">View MathML</a>

Now, we prove an elementary lemma.

Lemma 2.3. If g Ł1 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M24">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M25">View MathML</a>

Proof.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M26">View MathML</a>

Remark 2.4. Lemma 2.2 shows that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M27">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M28">View MathML</a>

Lemma 2.5. There exists a suitable constant C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M29">View MathML</a>

Proof.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M30">View MathML</a>

But

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M31">View MathML</a>

By using Lemma 2.1 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M32">View MathML</a>

Applying Lemmas 2.1 and 2.2 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M33">View MathML</a>

Finally, using Lemma 2.1, we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M34">View MathML</a>

This ends the proof of Lemma 2.5.

Theorem 2.6. Suppose that (2.3) hold for some positive integer N, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M35">View MathML</a>

as λ → ∞.

Proof. We integrate (1.5) over [a, x] and obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M36">View MathML</a>

In particular

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M37">View MathML</a>

and so,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M38">View MathML</a>

We need to prove that two last terms are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M39">View MathML</a> as λ → ∞. Applying Lemmas 2.2 and 2.4 we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M40">View MathML</a>

When N = 1, applying Lemma 2.5, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M41">View MathML</a>. Now By using Lemma 2.3 and induction we achieve that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/40/mathml/M42">View MathML</a> as λ → ∞.

Remark 2.7. By using the discussions of choice of f in [5], the condition (2.3) let us to consider q as the form q(x) ~ x-K where 1 ≤ K < 2.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author would like to thank Professor Grigori Rozenblum for useful comments.

References

  1. 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. 99(1-2), 51–70 (1984). Publisher Full Text OpenURL

  2. Atkinson, FV: Asymptotics of an eigenvalue problem involving an interior singularity. Research program proceedings ANL-87-26, 2, pp. 1–18. Argonne National Lab. Illinois (1988)

  3. Harris, BJ: Asymptotics of eigenvalues for regular Sturm-Liouville problems. J Math Anal Appl. 183, 25–36 (1994). Publisher Full Text OpenURL

  4. 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. 110(1-2), 63–71 (1988). Publisher Full Text OpenURL

  5. Harris, BJ, Race, D: Asymptotics of eigenvalues for Sturm-Liouville problems with an interior singularity. J Diff Equ. 116(1), 88–118 (1995). Publisher Full Text OpenURL

  6. Harris, BJ, Marzano, F: Eigenvalue approximations for linear periodic differential equations with a singularity. Electron J Qual Theory Diff Equ. 1–18 No. 7 (1999)

  7. Coskun, H, Bayram, N: Asymptotics of eigenvalues for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition. J Math Anal Appl. 306(2), 548–566 (2005). Publisher Full Text OpenURL

  8. Fix, G: Asymptotic eigenvalues of Sturm-Liouville systems. J Math Anal Appl. 19, 519–525 (1967). Publisher Full Text OpenURL

  9. Fulton, CT, Pruess, SA: Eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems. J Math Anal Appl. 188(1), 297–340 (1994). Publisher Full Text OpenURL

  10. Fulton, CT: Two point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc Roy Soc Edinburgh Sect A. 77, 293–308 (1977)