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On the Expansion Formula for a Class of Dirac

Operator with Discontinuous Coefficient

Kh. R. Mamedov and Aynur C¸ ¨

OL

Abstract— In this paper we consider a first order differential equation system with a discontinuous coefficient and spectral parameter dependent boundary condition in the half line. The operator interpretation of the given boundary value problem is investigated in the Hilbert space H = L2,ρ

¡

0, ∞; C× C. The resolvent operator is constructed and the expansion formula with respect to eigenfunctions is obtained. Copyright c° 2009 Yang’s Scientific Research Institute, LLC. All rights reserved.

Index Terms— Dirac operator, expansion formula, resolvent operator.

I. INTRODUCTION

W

E consider the boundary value problem

BY0+ Ω (x) Y = λρ (x) Y, 0 < x < ∞, (1) Y1(0) − λY2(0) = 0 (2) where B = Ã 0 1 −1 0 ! , Ω (x) = Ã p (x) q (x) q (x) −p (x) ! , Y = Ã Y1(x) Y2(x) ! ,

p (x), q (x) are real measurable functions, λ is a spectral

parameter and ρ (x) = ( α, 0 ≤ x ≤ a, 1, a < x < ∞, and 1 6= α > 0.

Assume that the condition

Z 0

kΩ(x)k dx < ∞ (3) is satisfied for Euclidean norm of the matrix function Ω(x).

The spectral analysis of the boundary-value problem (1)-(2) in the half line in the case of ρ (x) ≡ 1 in the equation (1) was researched in [5], [8], [7], [1], [12]. In finite interval the expansion formula for Dirac operator with a spectral parameter

Manuscript received June 23, 2009; revised August 15, 2009.

Kh. R. Mamedov and Aynur C¸ ¨OL, Mathematics Department, Science and Letters Faculty, Mersin University, 33343, Mersin-Turkey. Emails: hanlar@mersin.edu.tr(Kh. R. Mamedov), acol@mersin.edu.tr(Aynur C¸ ¨OL).

Publisher Item Identifier S 1542-5908(09)10403-7/$20.00

Copyright °2009c Yang’s Scientific Research Institute, LLC. All rights reserved. The online version posted on December 22, 2009 at http://www.YangSky.com/ijcc/ijcc74.htm

in the boundary conditions was examined in [11] and a dis-continuous Sturm-Liouville problem with a spectral parameter in boundary condition was investigated in [3], [10]. In the half line the spectral analysis of Sturm-Liouville operator was investigated in [4]. The discontinuous inverse problem of scattering theory for the system (1) with the boundary condition not containing a spectral parameter was studied in [6], [9]. The spectral properties of Dirac operators on (0, 1) with potentials that belong to entrywise to Lp(0, 1), for some p ∈ [1, ∞), were studied and the vast reference list about the

inverse problems for Dirac operators on the finite interval were given in [2].

In this paper our aim is to obtain the expansion formula for the boundary-value problem (1)-(2) in the half line. To obtain the resolvent operator and the expansion formula we applied the method [13]. Let µ (x) = ( a + α (x − a) , 0 ≤ x ≤ a, x, x > a.

It is easily shown that the vector function

f0(x, λ) = Ã 1 −i ! eiλµ(x)

is a solution of the equation (1) when Ω (x) ≡ 0.

As known from [6], when the condition (3) is satisfied, for Imλ ≥ 0 the equation (1) has an solution f (x, λ) which can be expressed uniquely as f (x, λ) = f0(x, λ) + Z µ(x) K (x, t) µ 1 −ieiλtdt (4) satisfying the condition

lim x→∞f (x, λ) e −iλx= Ã 1 −i ! .

Moreover, the elements of the matrix kernel K (x, t) are summable on the positive half line and for the Euclidean norm of K (x, t), the inequality Z µ(x) kK (x, t)k dt ≤ eσ(x)− 1 (5) is satisfied, where σ (x) =R x kΩ ((t))k dt.

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Let Y (x, λ) and Z (x, λ) be vector solutions of the equa-tions system (1). The expression

W [Y (x, λ) , Z (x, λ)] = YT(x, λ) BZ (x, λ) = (Y1, Y2) Ã 0 1 −1 0 ! Ã Z1 Z2 ! = Y1Z2− Y2Z1

is called Wronskian of the vector functions Y (x, λ) and

Z (x, λ) .

Since p (x) and q (x) are real valued functions, the vec-tor functions f (x, λ) and f (x, λ) are fundamental solutions system of the equation (1) for real λ. The Wronskian of the vector functions f (x, λ) and f (x, λ) doesn’t depend on x and is equal to 2i.

We denote by ϕ (x, λ) the solution of the equation (1) satisfying the conditions

ϕ1(0, λ) = λ, ϕ2(0, λ) = 1. Let us define the function

E (λ) ≡ f1(0, λ) − λf2(0, λ) .

It can be shown the function E (λ) hasn’t any zeros on the closed upper plane.

The paper is organized as follows: In Section II we give the theoretic formulation of operator of the boundary value prob-lem (1), (2) in the Hilbert space Hρ= L2,ρ

¡

0, ∞; C× C. In Section III we find the kernel for the resolvent operator and construct the resolvent operator. Finally, we obtain the expansion formula with respect to eigenfunctions in Section IV.

II. OPERATORTHEORETICFORMULATION

In the Hilbert space Hρ = L2,ρ ¡ 0, ∞; C× C an inner product is defined by (F, G) := Z 0 n f1(x) g1(x) + f2(x) g2(x) o ρ (x) dx + f3g3,

for the triple component vectors

F :=     f1(x) f2(x) f3     , G :=     g1(x) g2(x) g3     ∈ Hρ.

For convenience, we put

f := f (x) := Ã f1(x) f2(x) ! , therefore F = Ã f (x) f3 ! and l(y) := 1 ρ (x){BY 0+ Ω (x) Y } .

We define the operator L corresponding to the boundary value problem (1)-(2) as the following form

LF =     −f0 2(x) + p (x) f1(x) + q (x) f2(x) f0 1(x) + q (x) f1(x) − p (x) f2(x) f1(0)     , F ∈ D(L) where the domain of definition of the operator L is

D (L) := { F | F = (f1(x) , f2(x) , f3) ∈ Hρ, f1(x) , f2(x) are absolutely continuous in every [0, b] ⊂ [0, ∞) , l (f ) ∈ L2,ρ

¡

0, ∞; C, f3= f2(0)}

It is easy to show that the operator L with domain D (L) is selfadjoint in the space Hρ.

III. RESOLVENTOPERATOR

If λ is not a spectrum point of operator L, then there exists the resolvent Rλ= (L − λI)−1. Now we find this expression

of the operator Rλ.

Lemma III.1 The resolvent Rλ is the integral operator with the kernel which has the following form

Rλ(x, t) = − 1 E (λ) ( ϕ (x, λ) ef (t, λ) , x ≤ t, f (x, λ) eϕ (t, λ) , t ≤ x, here ef (t, λ) denotes the transposed vector function of f (t, λ) .

Proof: Let F ∈ D (L) and f (x) =

Ã

f1(x)

f2(x) !

be zero in exterior of every interval. To construct the resolvent operator of L, we need to solve the boundary value problem

BY0+ Ω (x) Y = λρ (x) Y + ρ (x) f (x) , (6) Y1(0) = λY2(0) + f3. (7) By applying the method of variation of parameters, we want to find the solution of the problem (6),(7) which has a form

Y (x, λ) = c1(x, λ) ϕ (x, λ) + c2(x, λ) f (x, λ) , (8) where ϕ (x, λ), f (x, λ) are solutions of homogeneous prob-lem. Then we get the equations system

c0 1(x, λ) ef (x, λ) Bϕ (x, λ) = f (x, λ) f (x) ρ (x) , (9)e c0 2(x, λ) eϕ (x, λ) Bf (x, λ) = ϕ (x, λ) f (x) ρ (x) .e Since Y (x, λ) ∈ L2,ρ ¡ 0, ∞; C, then c 1(∞, λ) = 0. Using this relation and the equation system (9) we get

c1(x, λ) = − Z x e f (t, λ) E (λ) f (t) ρ (t) dt, (10) c2(x, λ) = c2(0, λ) − x Z 0 e ϕ (t, λ) E (λ) f (t) ρ (t) dt.

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Substituting (10) into (8) we obtain Y (x, λ) = Z 0 Rλ(x, t) f (t) ρ (t) dt + c2(0, λ) f (x, λ) , where Rλ(x, t) = − 1 E (λ) ( ϕ (x, λ) ef (t, λ) , x ≤ t, f (x, λ) eϕ (t, λ) , t ≤ x. (11)

Taking formula (7) we get c2(0, λ) = E(λ)f3 . Thus

(L − λI)−1f = Z 0 Rλ(x, t) f (t) ρ (t) dt + f3 E (λ)f (x, λ) , f ∈ D (L) . (12) The lemma is proved.

Lemma III.2 Let the vector function f (x) be finite at infinity

and F = Ã f (x) f3 ! ∈ D (L) . Then Z 0 Rλ(x, t) f (t) ρ (t) dt + f3 E (λ)f (x, λ) (13) = −f (x) λ + f (x, λ) f1(0) λE (λ) + 1 λ Z 0 Rλ(x, t) g (t) dt, where g (x) = Bf0(x) + Ω (x) f (x) .

Proof: Using (11) and integrating by parts, we write

Eq. (14) at the top of the next page. This proves the lemma. Suppose that f (t) satisfies the condition of Lemma (III.2), then from Eq. (13) it follows that for Imλ > 0, |λ| → ∞,

Z 0 Rλ(x, t) f (t) ρ (t) dt+ f3 E (λ)f (x, λ) = − f (x) λ +O µ 1 λ, (15) since Z 0 Rλ(x, t) g (t) dt = O (1) .

The following lemma is well known. Lemma III.3 Rλ= Rλ.

IV. EXPANSIONFORMULA

With the help of these lemmas we obtain the expansion formula in eigenfunctions from D (L) of boundary value problem (1)-(2).

We integrate both sides of Eq. (15) with respect to λ over the circle ΓR of radius R and center at zero. As a result we

have − f (x) + 1 2πi I ΓR O µ 1 λ = 1 2πi I ΓR Z 0 Rλ(x, t) f (t) ρ (t) dt + 1 2πi I ΓR f3 E (λ)f (x, λ) dλ (16)

The function Rλ(x, t) is analytic in the upper and lower half

plane. Therefore, we have 1 2πi I ΓR Z 0 Rλ(x, t) f (t) ρ (t) dt = IR1 + IR2 + IR3, (17) where I1 R= 1 2πi R−iεZ −R−iε Z 0 Rλ(x, t) f (t) ρ (t) dt, (18) IR2 = 1 2πi −R+iεZ R+iε Z 0 Rλ(x, t) f (t) ρ (t) dt, (19) I3 R= 1 2πi    R+iεZ R−iε   Z 0 Rλ(x, t) f (t) ρ (t) dt dλ + −R−iεZ −R+iε   Z 0 Rλ(x, t) f (t) ρ (t) dt dλ   

and here ε is any positive number. From Lemma III.2 and Eq. (15) it follows that I3

R→ 0 as R → ∞. From Eq. (15) it

follows that the limit lim ε→0 Z 0 Rλ±iε(x, t) f (t) ρ (t) dt = Z 0 Rλ±i0(x, t) f (t) ρ (t) dt.

Therefore, by going over in Eq. (16) to the limit as R → ∞ and using Eqs. (17) - (19) we find Eq. (20) in the next page from Lemma III.3, Rλ−i0 = Rλ+i0. Let us compute Rλ+i0(x, t) − Rλ−i0(x, t) . Let ψ (x, λ) be solution of

equa-tion (1) with the initial condiequa-tions

ψ (0, λ) = Ã 0 1 ! . Then f (x, λ) = f2(0, λ) ϕ (x, λ) + E (λ) ψ (x, λ) . (21)

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Z 0 Rλ(x, t) f (t) ρ (t) dt = − 1 E (λ)f (x, λ) x Z 0 e ϕ (t, λ) f (t) ρ (t) dt − 1 E (λ)ϕ (x, λ) × Z x e f (t, λ) f (t) ρ (t) dt = −f (x, λ) λE (λ) x Z 0 ½ −∂ ∂tϕ (t, λ) B + ee ϕ (t, λ) Ω (t) ¾ f (t) dt − 1 λE (λ)ϕ (x, λ) Z x ½ −∂ ∂tf (t, λ) B + ee f (t, λ) Ω (t) ¾ f (t) dt = 1 λE (λ)f (x, λ) eϕ (t, λ) Bf (t) ¯ ¯ ¯ ¯ x t=0 + 1 λE (λ)ϕ (x, λ) ef (t, λ) Bf (t) ¯ ¯ ¯ ¯ t=x 1 λ Z 0 Rλ(x, t) g (t) dt = −f (x) λ f3 E (λ)f (x, λ) + 1 λ Z 0 Rλ(x, t) g (t) dt + f1(0) λE (λ)f (x, λ) (14) f (x) = − lim R→∞ 1 2πi I ΓR   Z 0 Rλ(x, t) f (t) ρ (t) dt dλ − 1 2πi I ΓR f3 E (λ)f (x, λ) dλ = − lim R→∞ © IR1 + IR2 + IR3 ª 1 2πi I ΓR f3 E (λ)f (x, λ) dλ (20) = − 1 2πi Z −∞ Z 0 [Rλ−i0(x, t) − Rλ+i0(x, t)] f (t) ρ (t) dt − 1 2πi Z −∞ · f (x, λ + i0) E (λ + i0) f (x, λ − i0) E (λ − i0) ¸ f3

Since ϕ (x, λ) and ψ (x, λ) are entire vector functions of λ, it follows from Eqs. (12) and (21) that

Z 0 [Rλ+i0(x, t) − Rλ−i0(x, t)] f (t) ρ (t) dt = − Z 0 £ Rλ+i0(x, t) − Rλ+i0(x, t) ¤ f (t) ρ (t) dt = x Z 0 " f (x, λ) E (λ) f (x, λ) E (λ) # e ϕ (t, λ) f (t) ρ (t) dt + Z x ϕ (x, λ) " e f (t, λ) E (λ) e f (t, λ) E (λ) # f (t) ρ (t) dt.

On the other hand

f (x, λ) E (λ) f (x, λ) E (λ) = ( f2(0, λ) E (λ) f2(0, λ) E (λ) ) ϕ (x, λ) = f1(0, λ)f2(0, λ) − f1(0, λ) f2(0, λ) |E (λ)|2 ϕ (x, λ) = −2i 1 |E (λ)|2ϕ (x, λ) . Therefore 1 2πi Z 0 {Rλ+i0(x, t) − Rλ−i0(x, t)} f (t) ρ (t) dt = −1 π Z 0 ϕ (x, λ) eϕ (t, λ) |E (λ)|2 f (t) ρ (t) dt.

We obtained the following expansion by eigenfunctions of the operator L on the form

f (x) = 1 π Z −∞ Z 0 ϕ (x, λ) eϕ (t, λ) |E (λ)|2 f (t) ρ (t) dt +1 π Z 0 ϕ (x, λ) |E (λ)|2f3dλ, f3 = 1 π Z −∞ Z 0 e ϕ (t, λ) |E (λ)|2f (t) ρ (t) dt +1 π Z 0 f3 |E (λ)|2dλ.

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or f (x) = 1 Z −∞ Z 0 u (x, λ) u∗(t, λ) f (t) ρ (t) dt + 1 2πi Z 0 u (x, λ) E (λ) f3dλ, f3 = 1 2πi Z −∞ Z 0 u∗(t, λ) E (λ) f (t) ρ (t) dt +1 π Z 0 f3 |E (λ)|2dλ, where u (x, λ) = f (x, λ) −E (λ) E (λ)f (x, λ) , u∗(x, λ) = f (x, λ) −e E (λ) E (λ)f (x, λ)

and u∗ denotes the conjugated vector function of eu.

REFERENCES

[1] Mark J. Ablowitz. Segur, harvey solitons and the inverse scattering transform. Society for Industrial and Applied Mathematics(SIAM) Studies in Applied Mathematics, pages x+425, 1981.

[2] S. Albeverio, R. Hryniv, and Ya. Mykytyuk. Inverse spectral problems for dirac operators with summable potentials. Russ. J. Math. Phys., 12(4):406–423, 2005.

[3] M. Demirci, Z. Akdo˘gan, and O. Sh. Mukhtarov. Asymptotic behavior of eigenvalues and eigenfunctions of one discontinuous boundary-value problem. International Journal of Computational Cognition, 2(3):101– 113, 2004.

[4] Charles T. Fulton. Singular eigenvalue problems with eigenvalue param-eter contained in the boundary conditions. Proc. Roy. Soc. Edinburgh

Sect. A, 87(1-2):1–34, 1980/81.

[5] M. G. Gasymov. An inverse problem of scattering theory for a system of dirac equations of order 2n. Trudy Moskov. Mat. Obˇsˇc, 19:41–112, 1968(Russian).

[6] I. M. Guse˘ınov. The inverse scattering problem for a system of dirac equations with discontinuous coefficients. Dokl. Akad. Nauk Azerba˘ıdzhana, 55(1-2):13–18, 1999(Russian).

[7] B. M. Levitan and I. S. Sargsjan. Introduction to spectral theory, Selfadjoint ordinary differential operators. Nauka, Moscow, 1970.

[8] B. M. Levitan and I. S. Sargsjan. Sturm-Liouville and Dirac operators. Mathematics and its Applications(Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991 (translated from the Russian). [9] Kh. R. Mamedov and Aynur C¸¨ol. On the inverse problemof the scattering

theory for a class of systems of dirac equations with discontinuous coefficient. Eur. J. Pure Appl. Math., 1(3):21–32, 2008.

[10] Khanlar R. Mamedov. On a basic problem for a second order differential equation with a discontinuous coefficient and a spectral parameter in the boundary conditions. In Geometry, integrability and quantization, pages 218–225, Softex, Sofia, 2006.

[11] S. G. Mamedov. The inverse boundary value problem on a finite interval for dirac’s system of equations. Azerba˘ıdˇzan. Gos. Univ. Uˇcen. Zap. Ser.

Fiz-Mat, (5):61–67, 1975.

[12] Vladimir A. Marchenko. Operator Theory: Advances and

Applica-tions, chapter Sturm-Liouville operators and applicaApplica-tions, pages xii+367.

Birkh¨auser Verlag, Basel, 1986(Translated from the Russian by A. Ia-cob).

[13] E. C. Titchmarsh. Eigenfunction expansions associated with

second-order differential equations. Clarendon Press, Oxford, second edition,

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