A Uniqueness the Theorem for Singular Sturm-Liouville Problem = Singüler Sturm-Liouville Problemi için Teklik Teoremi

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C.Ü. Fen-Edebiyat Fakültesi

Fen Bilimleri Dergisi (2003)Cilt 24 Sayı 2

A Uniqueness the Theorem for Singular Sturm-Liouville Problem

Etibar S. PANAKHOVand Hikmet KOYUNBAKAN

Firat University , Department of Mathematics 23119 Elazığ / TURKEY E-mail: epenahov@hotmail.com, hkb63@yahoo.com

Received:09.03.2004, Accepted: 16.03.2004

Abstract. In this paper, we show that If q(x) is prescribed on the

(

π 2,π

]

then the one spectrum suffices to determine q(x) on the interval

(

.The potential function q(x) in a Sturm Liouville problem is uniquely determined with one spectra by using the Hochstadt and Lieberman’s method [2].

)

0,π 2 Key Words: Sturm-Liouville problem, Spectrum

Singüler Sturm-Liouville Problemi için Teklik Teoremi

Özet: Bu makalede gösterdi ki q(x)

(

π 2,π

]

aralığında tanımlanmış ise

(

aralığı üzerinde q(x) fonksiyonunu belirlemek için bir spektrum yeterlidir. Sturm-Liouville probleminde q(x) potansiyel fonksiyonu Hochstadt ve Lieberman metodu kullanılarak bir spektruma göre tek olarak belirlenir.

)

0,π 2

Anahtar Kelimeler: Sturm-Liouville problem, Spectrum Introduction.

In this paper, we shall be concerned with an inverse Sturm-Liouville operator. We consider the operator

y y x x q y Ly ν _ =λ      ₊ − + ′′ − = ( ) 2 ₂1/4 (1)

(2)

with the boundary conditions ) 1 ( 2 1 ) , ( lim ₁_/₂ 0 − = Γ + → ν λ ν ν x x y x , (2) . (3) 0

]

}

sin ) , ( cos ) , (π λ β+ y′π λ β = y

The operator L is Self-Adjoint on the and with (2)–(3) boundary conditions has a discret spectrum

{

. If condition (3) is replaced by

[

0,π 2 L n λ . (4) 0 sin ) , ( cos ) , (π λ γ + y′π λ γ = y

So, we obtain a new spectrum

{ }

λ′n .

In this paper, we will consider a variation of the above inverse problem in that we will not require any information about a second spectrum but rather suppose q(x) is known almost everywhere on

    π _π , 2 .

This information together with the spectrum

{

of the problem (1)–(3) will be shown to determine q(x) uniquely on

(

.

}

n

λ

]

0,π

Theorem : We get the operator (1) with the boundary conditions (2) and (3). Let be the spectrum of L with (2) and (3). Consider a second operator

{ }

λn y y x x q y y L ν _ =λ      ₊ − + ′′ − = ~( ) 2 ₂1/4 ~ ₍₅₎

where q~

( )

x is summable on the interval

(

0,π

]

and

( )

x q x q( )=~ (6) on the interval     π _π , 2

( )

x

. Suppose that the spectrum of with the (2)–(3) is also . Then almost everywhere on .

L~

{ }

λ_n

q x

q( )=~

(

0,π

]

Proof : Before proving the theorem we will first mention some results which will be need later. We take the following problems

y y x x q y Ly ν _ =λ      ₊ − + ′′ − = ( ) 2 ₂1/4 (7) ) 1 ( 2 1 ) , ( lim ₁_/₂ 0 − = Γ + → ν λ ν ν x x y x (8)

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y y x x q y y L ν _ =λ      − + + ′′ − = ~( ) 2 ₂1/4 ~ (9) ) 1 ( 2 1 ) , ( ~ lim ₁_/₂ 0 − = Γ + → ν λ ν ν x x y x (10)

As known [6], the Bessel’s functions of the first kind of order ν is following asymptotic relations:

( )

            +     ₋ ₋ = x O x x x J 1 4 2 cos 2 νπ π π ν , (11)

( )

      +     ₋ ₋ − = ′ 1 4 2 sin 2 O x x x J νπ π π ν . (12)

It addition , It can be shown [5] that there exist a kernel H(x,t) continuous on such that every solution of (7) and (8) can be expressed in the form

[ ] [

0,π × 0,π

]

( )

x J

( )

x H x t

( )

t J

( )

tdt x y x

∫

+ = 0 ) , ( , λ λ λ λ λ _ν _ν _ν _ν (13)

Where the kernel H ,

(

x t

)

is solution of following problem

) , ( ) ( 4 / 1 ) , ( ) , ( 4 / 1 ) , ( 2 2 2 2 2 2 2 2 t x H t q t t t x H t x H x x t x H       + − + ∂ ∂ = − + ∂ ∂ ν ν , ) ( ) , ( 2 q x dx t x dH ₌ , . 0 ) 0 , (x = H

Analogous results to (13) hold for in terms of a kernel which has similar properties of the . Using equation (13) and Its for we find that

) , ( ~_y _x _λ _{H ,}~

₍

_x _t ) λ

)

(

x t H , ~ xy( ,

( )

(

)

( ) ( )

( )

(

) ( )

( )

(

)

2 2 2 0 0 0 , , , , . x x x xt x y y J x H x t H x t J x J t dt x x H x t J t dt H x s J s ds ν ν ν ν ν ν ν ν λ λ λ λ λ λ λ λ   = + _ + _ ×

∫

% % % ν λ +

)

(14)

If the range of and is extended respect to the second argument and some straightforward computations , we rewrite (14) as

(

x t

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( )

_            ₋ ₋             ₋ ₋ + = x x

_∫

x H≈ x d y y 0 2 1 cos2 ₂ ₄ , cos2 ₂ ₄ 2 1 ~ _λ νπ π _τ _λ_τ νπ π _τ λ ν , (15) where

( )

(

)

(

)

        + + − + − + − =

∫

−

∫

− + − ≈ ds s x H s x H ds s x H s x H x x H x x H t x H x x x x τ τ τ τ τ τ 2 , ~ ) , ( 2 , ~ ) , ( ) 2 , ( ~ ) 2 , ( 2 , 2 2 . (16)

Now, we define the function

β λ π β λ π λ Ω( )= y( , )cos +y′( , )sin . (17) The zeros of Ω are the eigenvalues of or subject to (2)-(3) and if the asymptotic results of y and are considered the is a entire function of order

( )

λ L

(

λ L~ y′ Ω

)

₂1_of . λ

If we multiply (7) by and (9) by y and subtract we obtain , after integration , y′

(

yy′− yy′

)

+

_∫

x

(

q−q

)

yydx= 0 0 ~ ~ 0 ~ ~ π . (18) Using (6) - (8) - (10) , we obtain

[

~

(

,

) (

' ,

) (

,

) (

~' ,

)

]

2

(

~

)

0 0 0 + − = −y y

_∫

q qdx y y π π λ π λ π λ π λ π . (19) Now , (20) q q Q= ~− and dx y y x Q K =

∫

2 0 ~ ) ( ) ( π λ . (21)

If the properties of y and are considered , the function is a entire function and for λ = , since the first term of (19) is zero ,

y~ K

( )

λ n λ . (22)

( )

_n =0 K λ

In addition using (13) and (21) for 0< x≤π ,

( )

ν

λ

λ) 1 ₂

( M

(5)

where M is constant. Now ,

( )

λ Ω λ λ Ψ( )= K , (24)

( )

λ

Ψ is a entire function. Asymptotic form of Ω(λ) and with (23)

( )

₁ 2 1 O ν λ λ +     Ψ =     . So , From the Liouville Theorem for all λ

( )

λ Ψ =0 (25) or 0 . (26) ) (λ = K

From now on , substituting (15) into (21)

( )

( , ) 2 2 4 0 ~~ 4 2 2 1 ) ( 2 1 0 2 2 0 =               ₋ ₋ +             ₋ ₋ +

_∫

∫

Q x x Cos x H x Cos d dx x τ π νπ τ λ τ π νπ λ λ ν π . (27)

This can be written as

( )

∫

( )

∫

 =         +       ₋ ₋ + 2 0 2 2 2 0 2 ( , ) 0 ~~ ) ( ) ( 4 2 2 ) ( π π τ ν π ν τ τ τ π νπ τ λ λ τ λ Q x dx Cos Q Q x H x dx d x .

Letting λ→∞ for real , we see from Riemann -Lebesque Lemma that we must have λ

∫

2 = 0 0 ) ( π dx x Q (29) and

∫

=           +       ₋ ₋ 2 0 2 0 ) , ( ~~ ) ( ) ( 4 2 2 π π τ τ τ τ π νπ τ λ Q Q x H x dx d Cos (30)

But from the completeness of the functions Cos , we see that

∫

= < < + 2 2 0 , 0 ) , ( ~~ ) ( ) ( π τ π τ τ τ Q x H x dx Q (31)

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) ( ~ ) (x q x q = almost everywhere. References

1. Gasymov, M.G., The Definition of Sturm-Liouville Operator from Two Spectra, DAN SSSR, Vol.161, No:2, 1965, 274-276

2. Hochstadt , H. and Lieberman , B., An Invers Sturm Liouville Problem with Mixed Given Data. Siam J. Appl. Math. , Vol.34 , No: 4 , 1978, 676-680

3. Levitan, B.M., On the Determination of the Sturm - Liouville Operator from One and Two Spectra, Math. USSR Izvestija, vol. 12 , 1978, 179-193

4. Panakhov , E . S. , The definition of differential operator with peculiarity in zero on two spectrum. VINITI , N 4407 - 8091980 , 1980 , 1-16

5. Volk, V.Y., On Inverse Formulas for a Differential Equation with a Singularity at

x=0, Usp.Mat.Nauk (N.S.) 8(56), 1953, 141-151

6. Watson, G., A Treatise on the Theory of Bessel Functions, Cambridge University