Open Access

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

Boundary Value Problems20122012:40

DOI: 10.1186/1687-2770-2012-40

Received: 19 October 2011

Accepted: 12 April 2012

Published: 12 April 2012

Abstract

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

- y ( x ) + q ( x ) y ( x ) = λ y ( x ) , x ( a , b ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equa_HTML.gif

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
- y ( x ) + q ( x ) y ( x ) = λ y ( x ) , x ( a , b ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ1_HTML.gif
(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
y ( a ) cos α - y ( a ) sin α = 0 , α [ 0 , π ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ2_HTML.gif
(1.2)
y ( b ) cos β - y ( b ) sin β = 0 , β [ 0 , π ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ3_HTML.gif
(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-kwith 1 k < 4 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_IEq1_HTML.gif and 1 k < 3 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_IEq2_HTML.gif 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 [710]. We consider q(x) = Cx-Kwhere 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 [25]. 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:
y [ 0 ] : = y , y [ 1 ] : = y + f y , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equb_HTML.gif
y is a solution of (1.1) with boundary conditions (1.2) and (1.3) if and only if
y [ 0 ] y [ 1 ] = - f 1 f + q - f 2 - λ f y [ 0 ] y [ 1 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ4_HTML.gif
(1.4)
The object of the regularization process is to chose f in such way that
f L 1 ( a , b ) and - F : = q - f 2 + f L 1 ( a , b ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ5_HTML.gif
(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
tan θ = λ 1 2 y y [ 1 ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equc_HTML.gif
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
θ = λ 1 2 - f sin ( 2 θ ) + λ - 1 2 F sin 2 ( θ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ6_HTML.gif
(1.6)
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:
in Case 1 ( α = 0 , β = 0 ) : θ ( b , λ ) - θ ( a , λ ) = ( n + 1 ) π ; in Case 2 ( α = 0 , β 0 ) : θ ( b , λ ) - θ ( a , λ ) = ( n + 1 2 ) π - λ - 1 2 cot β + O λ - 3 2 ; in Case 3 ( α 0 , β = 0 ) : θ ( b , λ ) - θ ( a , λ ) = ( n + 1 2 ) π + λ - 1 2 cot α + O λ - 3 2 ; in Case 4 ( α 0 , β 0 ) : θ ( b , λ ) - θ ( a , λ ) = n π + λ - 1 2 ( cot α - cot β ) + O λ - 3 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equd_HTML.gif
It follows from (1.5-1.6) that large positive eigenvalues of either the Dirichlet or non-Dirichlet problems over [a, b] satisfy
λ 1 2 = θ ( b ) - θ ( a ) ( b - a ) + O ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ7_HTML.gif
(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 o λ - N 2 ( N 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_IEq3_HTML.gif. 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 ξ j (t) for j = 1, ..., N + 1, t [a, b] by
ξ 1 ( t ) : = 0 t f ( s ) + F ( s ) d s ξ j ( t ) : = 0 t ( f ( s ) + F ( s ) ) ξ j - 1 ( s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ8_HTML.gif
(2.1)
and note that in view of f, F L(a, b),
ξ j ( t ) c ξ j - 1 ( t ) for t [ a , b ] , 2 j N + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ9_HTML.gif
(2.2)
Suppose that for some N ≥ 1,
f ξ N + 1 , f 2 ξ N , f F ξ N L [ a , b ] ; f ( t ) ξ N + 1 ( t ) 0 as t 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ10_HTML.gif
(2.3)
We define a sequence of approximating functions a
θ 0 ( x ) : = θ ( a ) + λ 1 2 ( x - a ) ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ11_HTML.gif
(2.4)
θ j ( 0 ) : = θ ( 0 ) ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ12_HTML.gif
(2.5)
θ j + 1 ( x ) : = θ ( a ) + λ 1 2 ( x - a ) - a x f sin ( 2 θ j ( t ) ) d t + λ - 1 2 a x F sin 2 ( θ j ( t ) ) d t . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ13_HTML.gif
(2.6)
for j = 0, 1, 2, ... and for axb. We measure the closeness of the approximation in the next result. Thus
θ j + 1 = λ 1 2 - f sin ( 2 θ j ) + λ - 1 2 F sin 2 ( θ j ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equ14_HTML.gif
(2.7)

The following lemma appears in [2, 5].

Lemma 2.1. If g Ł1then for any j and axb
a x g ( t ) sin ( 2 θ j ( t ) ) d t = o ( 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Eque_HTML.gif

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
θ j + 1 - θ j C sup a x b θ - θ j ξ j + 1 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equf_HTML.gif

Now, we prove an elementary lemma.

Lemma 2.3. If g Ł1and θ ( x ) - θ j ( x ) = λ - 1 2 a x g { sin 2 ( θ ( t ) ) - sin 2 ( θ j ( t ) ) } d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_IEq4_HTML.gifthen θ ( x ) - θ j + 1 ( x ) λ - 1 2 sup a x b θ ( x ) - θ j ( x ) a x g d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_IEq5_HTML.gif

Proof.
θ ( x ) θ j + 1 ( x ) = λ 1 2 a x g { sin 2 ( θ ( t ) ) sin 2 ( θ j ( t ) ) } d t = 1 2 λ 1 2 a x g { cos ( 2 θ j ( t ) ) cos ( 2 θ ( t ) ) } d t = λ 1 2 a x g sin ( θ j ( t ) θ ( t ) ) sin ( θ j ( t ) + θ ( t ) ) } d t λ 1 2 sup a x b | θ ( x ) θ j ( x ) | a x g d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equg_HTML.gif

Remark 2.4. Lemma 2.2 shows that if θ ( x ) - θ j ( x ) = o λ - j 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_IEq6_HTML.gifthen θ ( x ) - θ j + 1 ( x ) = o λ - ( j + 1 ) 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_IEq7_HTML.gif

Lemma 2.5. There exists a suitable constant C such that
a x f ( sin ( 2 θ j ( t ) ) - sin ( 2 θ ( t ) ) ) d t C λ - 1 2 sup a x b θ ( x ) - θ j ( x ) , x ( a , b ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equh_HTML.gif
Proof.
a x f ( sin ( 2 θ ( t ) ) ( 1 ) - sin ( 2 θ j ( t ) ) ( 1 ) ) d t = λ - 1 2 a x f { sin ( 2 θ ) θ - sin ( 2 θ j ) θ j } d t + λ - 1 2 a x f 2 { sin 2 ( 2 θ ) - sin ( 2 θ j ) sin ( 2 θ j - 1 ) } d t - λ - 1 a x f F { sin ( 2 θ ) sin 2 ( θ ) - sin ( 2 θ j ) sin 2 ( θ j - 1 ) } d t = : I 1 + I 2 - I 3 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equi_HTML.gif
But
I 1 = λ - 1 2 [ f ( t ) ( sin 2 ( θ ( t ) ) - sin 2 ( θ j ( t ) ) ) ] a x - λ - 1 2 a x f ( t ) { sin 2 ( θ ) - sin 2 ( θ j ) } d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equj_HTML.gif
By using Lemma 2.1 we have
I 1 C 1 λ - 1 2 sup a x b θ ( x ) - θ j ( x ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equk_HTML.gif
Applying Lemmas 2.1 and 2.2 we have
I 2 : = λ - 1 2 a x f 2 ( t ) { sin ( 2 θ ) - sin ( 2 θ j ) } sin ( 2 θ ) d t + λ - 1 2 a x f 2 ( t ) { sin ( 2 θ ) - sin ( 2 θ j ) } sin ( 2 θ j ) d t + λ - 1 2 a x f 2 ( t ) { sin ( 2 θ j ) - sin ( 2 θ j - 1 ) } sin ( 2 θ j ) d t C 2 λ - 1 2 sup a x b θ ( x ) - θ j ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equl_HTML.gif
Finally, using Lemma 2.1, we conclude
I 3 : = λ - 1 a x f F { sin ( 2 θ ) - sin ( 2 θ j ) } sin 2 ( θ ) d t + λ - 1 a x f F ( sin ( θ ) - sin ( θ j - 1 ) ) ( sin ( θ ) + sin ( θ j - 1 ) ) sin ( 2 θ j ) d t C 3 λ - 1 2 sup a x b θ ( x ) - θ j ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equm_HTML.gif

This ends the proof of Lemma 2.5.

Theorem 2.6. Suppose that (2.3) hold for some positive integer N, then
θ ( b ) - θ ( a ) - ( b - a ) λ 1 2 = - a b f sin ( 2 θ N ( x ) ) d x + λ - 1 2 a b F sin 2 ( θ N ) d x + o λ - N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equn_HTML.gif

as λ → ∞.

Proof. We integrate (1.5) over [a, x] and obtain
θ ( x ) - θ ( a ) = λ 1 2 ( x - a ) - a x f sin ( 2 θ ( t ) ) d t + λ - 1 2 a x F sin 2 ( θ ( t ) ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equo_HTML.gif
In particular
θ ( b ) - θ ( a ) = λ 1 2 ( b - a ) - a b f sin ( 2 θ ( t ) ) d t + λ - 1 2 a b F sin 2 ( θ ( t ) ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equp_HTML.gif
and so,
θ ( b ) - θ ( a ) - ( b - a ) λ 1 2 = - a b f sin ( 2 θ N ( x ) ) d x + λ - 1 2 a b F sin 2 ( θ N ) d x + a b f { sin ( 2 θ N ( x ) - sin ( 2 θ ( x ) ) ) } d x + λ - 1 2 a b F { sin 2 ( θ ) - sin 2 ( θ N ) } d x . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equq_HTML.gif
We need to prove that two last terms are o λ - N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_IEq8_HTML.gif as λ → ∞. Applying Lemmas 2.2 and 2.4 we have
I : = a b f ( x ) { sin ( 2 θ N ( x ) - sin ( 2 θ ( x ) ) ) } d x + λ - 1 2 a b F ( x ) { sin 2 ( θ ) - sin 2 ( θ N ) } d x C λ - 1 2 sup a x b θ ( x ) - θ N ( x ) + C λ - 1 2 a b F sup a x b θ ( x ) - θ N ( x ) d x https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_Equr_HTML.gif

When N = 1, applying Lemma 2.5, θ ( x ) - θ 1 ( x ) = o λ - j 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_IEq9_HTML.gif. Now By using Lemma 2.3 and induction we achieve that I = o λ - N 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-40/MediaObjects/13661_2011_Article_161_IEq10_HTML.gif 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-Kwhere 1 ≤ K < 2.

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematical Sciences, Division of Mathematics, Chalmers University of Technology and University of Gothenburg

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© Hormozi; licensee Springer. 2012

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