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.
Keywords:Sturm-Liouville equation; boundary condition; Prüfer transformation.
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 , an eigenvalue problem may be associate with (1.1) by imposing the boundary conditions
In , 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 , 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 and in [2,4] respectively. Harris and Race  generalized those results for the case 1 ≤ k < 2. In , 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 [, 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:
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
When y = 0, θ is defined by continuity [, 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  noticed that the function θ satisfies the differential equation
Let λ > 0 and the n-th eigenvalue λn of (1.1-1.3), then according to [, 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 → ∞.
2. Statement of result
We define a sequence ξj(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
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  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
Remark 2.4. Lemma 2.2 shows that if then
Lemma 2.5. There exists a suitable constant C such that
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
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 , 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.
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