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

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.

Consider the linear differential equation

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

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 ∈ L^l[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 and 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 ∈ L_1,Loc(U). As Harris did in [[5], p. 90], suppose that there exists some real function f on [a, 0) ∪ (0, b] in AC_Loc([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:

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

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

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

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

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

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

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 → ∞.

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

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

Suppose that for some N ≥ 1,

We define a sequence of approximating functions a

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

The following lemma appears in [2,5].

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

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

Now, we prove an elementary lemma.

Lemma 2.3. If g ∈ Ł¹ and then

Proof.

Remark 2.4. Lemma 2.2 shows that if then

Lemma 2.5. There exists a suitable constant C such that

Proof.

But

By using Lemma 2.1 we have

Applying Lemmas 2.1 and 2.2 we have

Finally, using Lemma 2.1, we conclude

This ends the proof of Lemma 2.5.

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

as λ → ∞.

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

In particular

and so,

We need to prove that two last terms are as λ → ∞. Applying Lemmas 2.2 and 2.4 we have

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 [5], the condition (2.3) let us to consider q as the form q(x) ~ x^-K where 1 ≤ K < 2.

The author declares that they have no competing interests.

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

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

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

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

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

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

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

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

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)