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

Abstract

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

[Equation not available: see fulltext.]

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.[PUBLICATION ABSTRACT]

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

Correspondence: mailto:[email protected]

Web End [email protected] Department of Mathematical Sciences, Division of Mathematics, Chalmers University of Technology and University of Gothenburg, Gothenburg 41296, Sweden

RESEARCH Open Access

Mahdi Hormozi

Abstract

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

y (x) + q(x)y(x) = y(x), x (a, b)

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, Prfer transformation.

1. Introduction

Consider the linear differential equation

y (x) + q(x)y(x) = y(x), x (a, b), (1:1)

where l 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 y (a) sin = 0, [0, ), (1:2)

y(b) cos y (b) sin = 0, [0, ). (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 Prfer 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

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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 ax-k with 1 k < 43 and 1 k < 32 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:

y[0] := y, y[1] := y + fy,

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

y[0]y[1] = f 1f + q f 2 f y[0]y[1] (1:4)

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

f L1(a, b) and F := q f 2 + f L1(a, b). (1:5)

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

tan =

1

2 y

y[1] .

When y[1] = 0, 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 satisfies the differential equation

=

in Case 1 ( = 0, = 0) : (b, ) (a, ) = (n + 1);in Case 2 ( = 0, = 0) : (b, ) (a, ) = (n + 12)

in Case 3 ( = 0, = 0) : (b, ) (a, ) = (n + 12) +

in Case 4 ( = 0, = 0) : (b, ) (a, ) = n +

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

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1

2

f sin(2) +

1

2 Fsin2(). (1:6)

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

1

2 cot + O

1

2 cot + O

3 2

;

3 2

;

1

2 (cot cot ) + O

3 2

.

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1

2 = (b) (a)

(b a)

+ O(1). (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

N 2

(N 1) . To achieve this we first use the differential Equation (1.6) to obtain

estimates for (b) - (a) for general l as l .

2. Statement of result

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

1(t) :=

t

0

f (s) |+| F(s) |ds|

j(t) :=

0

t

(

f

(s)

+

j1(s)ds

(2:1)

F(s) )

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

j(t) cj1(t) for t [a, b], 2 j N + 1 (2:2)

Suppose that for some N 1,

f N+1, f 2N, fFN L[a, b];f (t)N+1(t) 0 as t 0.

(2:3)

We define a sequence of approximating functions a

0(x) := (a) +

1

2 (x a);

(2:4)

j(0) := (0); (2:5)

j+1(x) := (a) +

1

2 (x a)

x

x

a

f sin(2j(t))dt +

1

2

a

Fsin2(j(t))dt. (2:6)

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

j+1 =

1

2

f sin(2j) +

1

2 Fsin2(j) (2:7)

The following lemma appears in [2,5].

Lemma 2.1. If g 1 then for any j and a x b

x

a

g(t) sin(2j(t))dt = o(1)

as l .

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

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Lemma 2.2. There exists a suitable constant C such that

a

j+1

j

C sup

axb

j

j+1(x)

Now, we prove an elementary lemma.

Lemma 2.3. If g 1 and (x) j(x) =

1

2

xa g{sin2((t)) sin2(j(t))}dt

then

(x) j+1(x)

1

2 supaxb

(x) j(x)

xa gdt

Proof.

(x) j+1(x) =

x

1

2

a

g{sin2((t)) sin2(j(t))}dt

= 1

2

x

1

2

a

g{cos(2j(t)) cos(2(t))}dt

=

a

x

1

2

g sin(j(t) (t)) sin(j(t) + (t))}dt

1

2 sup

axb

(x) j(x)

x

a

gdt

Remark 2.4. Lemma 2.2 shows that if

(x) j(x)

= o

j 2

then

Lemma 2.5. There exists a suitable constant C such that

x

a

f (sin(2j(t)) sin(2(t)))dt C

(x) j+1(x)

= o

(j+1) 2

1

2 sup

axb

(x) j(x)

, x (a, b),

Proof.

x

a

x

f (sin(2(t))(1) sin(2j(t))(1))dt =

1

2

a

f {sin(2) sin(2j) j}dt

+

x

1

2

a

f 2{sin2(2) sin(2j) sin(2j1)}dt

1

x

fF{sin(2)sin2() sin(2j)sin2(j1)}dt

=: I1 + I2 I3.

But

a

x

I1 =

1

2 [f (t)(sin2((t)) sin2(j(t)))]xa

1

2

a

f (t){sin2() sin2(j)}dt

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By using Lemma 2.1 we have

I1 C1

1

2 sup

axb

(x) j(x)

.

Applying Lemmas 2.1 and 2.2 we have

I2 : =

1

2

a

x

f 2(t){sin(2) sin(2j)} sin(2)dt

+

x

1

2

a

f 2(t){sin(2) sin(2j)} sin(2j)dt

+

x

1

2

a

f 2(t){sin(2j) sin(2j1)} sin(2j)dt

C2

(x) j(x)

Finally, using Lemma 2.1, we conclude

I3 : = 1

x

1

2 sup

axb

a

fF{sin(2) sin(2j)}sin2()dt

+ 1

x

a

fF(sin() sin(j1))(sin() + sin(j1)) sin(2j)dt

C3

1

2 sup

axb

(x) j(x)

This ends the proof of Lemma 2.5.

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

(b) (a) (b a)

1

2 =

b

a

b

f sin(2N(x))dx +

1 2

a

Fsin2(N)dx + o

N 2

as l .

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

(x) (a) =

1

2 (x a)

x

a

a

x

f sin(2(t))dt +

1

2

Fsin2((t))dt

In particular

(b) (a) =

1

2 (b a)

b

a

b

f sin(2(t))dt +

1

2

a

Fsin2((t))dt

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and so,

(b) (a) (b a)

b

1 2 =

b

a

f sin(2N(x))dx +

1 2

a

Fsin2(N)dx

b

a

+ f {sin(2N(x) sin(2(x)))}dx

b

+

1 2

a

F{sin2() sin2(N)}dx.

We need to prove that two last terms are o

N 2

as l . Applying Lemmas 2.2

and 2.4 we have

I : =

b

b

a

f (x){sin(2N(x) sin(2(x)))}dx +

1 2

a

F(x){sin2() sin2(N)}dx

C

b

1

2 sup

axb

(x)

N(x)

+ C

1 2

a

F sup

axb

(x)

N(x)

dx

When N = 1, applying Lemma 2.5,

(x) 1(x)

= o

j 2

. Now By using Lemma

as l .

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.

2.3 and induction we achieve that I = o

N 2

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

Competing interestsThe author declares that they have no competing interests.

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

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2. Atkinson, FV: Asymptotics of an eigenvalue problem involving an interior singularity. Research program proceedings ANL-87-26, 2. pp. 118.Argonne National Lab. Illinois (1988)

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8. Fix, G: Asymptotic eigenvalues of Sturm-Liouville systems. J Math Anal Appl. 19, 519525 (1967). doi:10.1016/0022-247X(67)90009-19. Fulton, CT, Pruess, SA: Eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems. J Math Anal Appl. 188(1):297340 (1994). doi:10.1006/jmaa.1994.1429

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10. Fulton, CT: Two point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc Roy Soc Edinburgh Sect A. 77, 293308 (1977)

doi:10.1186/1687-2770-2012-40

Cite this article as: Hormozi: A note on a paper of Harris concerning the asymptotic approximation to the eigenvalues of -y + qy = ly, with boundary conditions of general form. Boundary Value Problems 2012 2012:40.

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