Research Article A Half-Inverse Problem for Impulsive Dirac Operator with Discontinuous Coefficient

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1 Abstract and Applied Analysis Volume 213, Article ID 18189, 4 pages Research Article A Half-Inverse Problem for Impulsive Dirac Operator with Discontinuous Coefficient YalçJn Güldü Cumhuriyet University, Faculty of Science, Department of Mathematics, 5814 Sivas, Turkey Correspondence should be addressed to Yalçın Güldü; yguldu@gmail.com Received 3 June 213; Accepted 9 September 213 Academic Editor: Agacik Zafer Copyright 213 Yalçın Güldü. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. An inverse problem for Dirac differential operators with discontinuity conditions and discontinuous coefficient is studied. It is shown by Hochstadt and Lieberman s method that if the potential function p() in Ω() is prescribed over the interval (, ), then a single spectrum suffices to determine p() on the interval (, ) and it is also shown here that ρ() is uniquely determined by a spectrum. 1. Introduction In this paper, we are concerned with the Dirac operator L generated by equation with l(y):=by () +Ω() y () =λρ() y (), I:=(, (1) 2 ) ( 2,) B=( 1 () q () ), Ω() =(p 1 q () p () ), ρ () = 1, < 2 α, 2 <, where 1<α R +, y() = ( y 1() y 2 () ),subjecttotheboundary conditions and discontinuity conditions (2) y 1 () =, (3) y 2 () =, (4) y( 2 +)=Ay( ), (5) 2 where p() and q() are real valued functions in L 2 (, ), λ is a spectral parameter, and A=( β β ), 1 β R+ \1}. Inverse problems of spectral analysis implicate the reconstruction of a linear operator from its spectral data (e.g., see [1 5]). Half inverse problem for a Dirac operator consists in reconstruction of this operator from its spectrum and half of the potential. The first result on the half-inverse problem is due to Hochstadt and Lieberman [6].Afterthat,Hald[7] proved that if the potential is known over half of the interval and if one boundary conditions is given, then the potential and the other boundary condition are uniquely determined by the eigenvalues. In [8, 9], Malamud and Gesztesy and Simon obtained some new uniqueness results in inverse spectral analysis with partial information on the potential for scalar and matri Sturm-Liouville equations, respectively. In 21, Sakhnovich [1] studied the eistence of solution to the halfinverse problem. In [11], necessary and sufficient condition for solvability of the half-inverse spectral problem for Sturm- Liouville operators with singular operator was taken. In [12], by using the Hochstadt and Lieberman s method, halfinverse problem was solved for diffusion operators. In [13], the authors presented half-inverse problem for the Sturm- Liouville equation with eigenparameter-dependent boundary conditions. On the other hand, the fundamental and detailed results about Dirac operators were given in [14]. Moreover, direct or

2 2 Abstract and Applied Analysis inverse spectral problems for Dirac operators are largely well studied in [1, 8, 15]. Therearealsosomestudiesabouttheinteriorinverse problems. Arutyunyan [16] provedthattheeigenvaluesλ n, n =,1,... and normalizing coefficients α n = y n L 2 (.1)} 2, n =,1,... uniquely determined the potential Q(). Malamud [8] proved an analog of Borg theorem [17]; he showed that the spectra of two boundary value problems for an operator with different boundary conditions at one end uniquely determined the potential Q(). He also proved an analog of the theorem of Hochstadt and Lieberman [6]; one spectrum and a potential on the interval (, 1/2) uniquely determined the potential Q() on the whole interval [, 1]. Ontheotherhand, in21, DelRioandGrébert [18]proved that in the case where φ is a priori known on [a, 1],thenonlya part (depending on a) of two spectra determined φ on [, 1]. Furthermore, inverse problems for interior spectral data of the Sturm-Liouville and Dirac operators were studied in [19 22]. The jump conditions like (5) appear in-some important physical problems. The work in [7] is a well-known work about discontinuous inverse eigenvalue problems. Direct and inverse problems for Dirac operators with discontinuities inside the interval were investigated in [23]. In this paper, by using the Hochstadt and Lieberman s method in [6], we discuss the half-inverse problem for Dirac operator with discontinuity conditions and discontinuous coefficients (1) (5). Furthermore, the potential p() and discontinuous coefficient ρ() are uniquely determined. 2. Statement of Results Let the function φ(, λ) : I R 2 be solution of (1) which satisfies the initial conditions φ (, λ) =( 1 ) (6) and the jump conditions (5). It is shown in [14] that,forthesolutionφ(, λ), the following representation holds: φ (, λ)=φ (, λ) + K (, t) φ (t, λ) dt, for << 2, (7) β ± = (1/2)(β± β), K(, t) = (K ij (, t)) i,j=1,2,andk ij (, t) are real valued continuous functions for i, j = 1, 2 and for each, μ () =, α α() + (), Net, we define the function < <. (1) Δ (λ) =φ 2 (, λ). (11) The zeros of Δ(λ) which is called the characteristic function of the problem (1) (5) are the eigenvalues of L. From the equalities (7) (11), we have Δ (λ) =Δ (λ) +o(ep μ ()), (12) where Δ (λ) = β + cos λμ() + β cos λ( μ()) and = Im λ. Theorem 1. (i) The problem L has denumerable many eigenvalues such that all of them are real and simple. (ii) The eigenvalues λ n are epressed by the following asymptotic formula: λ n =λ n +O(1), (13) where λ n are the zeros of Δ (λ) and λ n sup n h n <. = n/μ() + h n, Proof. (i) Since Δ(λ) is entire function, it has denumerable many zeros. Moreover, from [23], zeros of λ n } are real and simple. = n/μ() + O(1). Itis (ii)itisshownin[24] thatλ n obvious that Δ (λ) C δ ep μ() for λ G δ := λ : λ λ n >δ}and Δ(λ) Δ (λ) = o(ep μ()). Therefore, it follows from the Rouche s theorem that the functions Δ (λ) and Δ(λ) havethesamenumberofzeros inside the contour γ ε := λ : λ = λ n ε};thatis,the eigenvalues λ n aregivenbythefollowingasymptoticformula: λ n = n μ () (1 + O ( 1 )). (14) n Consider a second operator L generated by the differential equation where φ (, λ) = (φ 1 (, λ), φ 2 (, λ)) T has the form φ 1 (, λ) = φ 2 (, λ) = sin λ, < 2 β + sin λμ () +β sin λ( μ()), 2 <, cos λ, < 2 β + cos λμ () +β cos λ( μ()), 2 <, (8) (9) By () + Ω () y () =λ ρ () y (), I:=(, (15) 2 ) ( 2,) subject to the same boundary conditions (3) and(4) and discontinuity condition (5). Here, Ω() = ( p() q() q() p() ) with a real valued function p() L 2 (, ). We denote eigenvalues by λ n and the corresponding eigenfunctions by φ n () = φ(, λ n ) of the problem L and denote eigenvalues by λ n and the corresponding eigenfunctions by φ n () = φ(, λ n ) of the problem L.

3 Abstract and Applied Analysis 3 Lemma 2. If λ n = λ n,thenα= α that is, ρ() = ρ(). Proof. Since λ n = (n/μ())(1 + O(1/n)), then λ n = (n/ μ())(1+o(1/n)), (n/μ())(1+o(1/n)) = (n/ μ())(1+ O(1/n)). Lettingn,then weconcludethatμ() = μ(). Moreover,sinceμ() = ()(α + 1), thenα= α. So ρ() = ρ(). Theorem 3. If λ n = λ n,foralln N and p() = p(), for (, ),thenp() = p() almost everywhere on (, ). Proof. Let us write (1)forthesolutionsφ and φ and take into account Lemma 2 as Bφ (, λ) +Ω() φ (, λ) =λρ() φ (, λ), B φ (, λ) + Ω () φ (, λ) =λρ() φ (, λ). (16) If we multiply these equalities by φ(, λ) and φ(, λ), respectively, and subtract, then we obtain d d φ 1 (, λ) φ 2 (, λ) φ 1 (, λ) φ 2 (, λ)} =[Ω() Ω ()]φ(, λ) φ (, λ). (17) Integrating the last equality from to with respect to, the equality φ 1 (, λ) φ 2 (, λ) φ 1 (, λ) φ 2 (, λ)} = [Ω () Ω ()]φ(, λ) φ (, λ) d = [p () p ()]Jφ(, λ) φ (, λ) d (18) is obtained where J:=( 1 1 ). Applying the initial condition (6) and the assumption p() = p(), (, ) in hypothesis, we get φ 1 (, λ) φ 2 (, λ) φ 1 (, λ) φ 2 (, λ)} = [Ω () Ω ()]φ(, λ) φ (, λ) d = [p () p ()]Jφ(, λ) φ (, λ) d. (19) Define F (λ) := [p () p ()]Jφ(, λ) φ (, λ) d, (2) where Jφ (, λ) φ (, λ) = cos 2λ + K 1 (, t) cos 2λtdt (21) + K 2 (, t) sin 2λtdt and K i (, t), i=1,2depend only on, t. Then we get from the boundary condition (4)that for all n. Now, define F(λ n )= (22) F (λ) χ (λ) := Δ (λ). (23) χ(λ) is an entire function. Since F(λ) = O(ep ) and Δ(λ) C δ ep μ() for λ G δ := λ : λ λ n > δ} where μ() = ()(α + 1), thenχ(λ) is constant from the Liouville s theorem. Furthermore, lim χ (λ) = λ λ R (24) from the equalities (7), (9), and (21) andtheriemann- Lebesgue lemma. Thus, χ(λ) = on the whole λ-plane. It follows from (2)and(21)that Q () cos 2λ K 1 (, t) cos 2λtdt K 2 (, t) sin 2λtdt} d = (25) for all λ where Q() := [p() p()]. Thiscanberewritten as cos 2λ [Q () + Q () K 1 (, t) d] dt + sin 2λt Q () K 2 (, t) d dt =. (26) From the completeness of the functions (cos 2λ, sin 2λt) T in L 2 (, ) L 2 (, ),wehave Q () + Q () K 1 (, t) d=, for << 2. (27) It follows that Q() = ; thatis,p() = p() almost everywhere for (, ). References [1] I. M. Gelfand and B. M. Levitan, On the determination of a differential equation from its spectral function, Izvestiya Akademii Nauk SSSR,vol.15,no.4,pp.39 36,1951. [2] B. M. Levitan, On the determination of the Sturm-Liouville operator from one and two spectra, Mathematics of the USSR- Izvestiya,vol.12,pp ,1978. [3] B. M. Levitan, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, The Netherlands, [4] J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Orlando, Fla, USA, [5]D.AlpayandI.Gohberg, Inverseproblemsassociatedtoa canonical differential system, in Recent Operator Theory and Related Topics,vol.127ofOperator Theory: Advances and Applications,pp.1 27,21.

4 4 Abstract and Applied Analysis [6] H. Hochstadt and B. Lieberman, An inverse Sturm-Liouville problemwithmiedgivendata, SIAM Journal on Applied Mathematics, vol. 34, no. 4, pp , [7] O. H. Hald, Discontinuous inverse eigenvalue problems, Communications on Pure and Applied Mathematics, vol.37,no. 5, pp , [8] M. M. Malamud, Uniqueness questions in inverse problems for systems of differential equations on a finite interval, TransactionsoftheMoscowMathematicalSociety,vol.6,pp , [9] F. Gesztesy and B. Simon, Inverse spectral analysis with partial information on the potential. II. The case of discrete spectrum, Transactions of the American Mathematical Society,vol.352,no. 6, pp , 2. [1] L. Sakhnovich, Half-inverse problems on the finite interval, Inverse Problems,vol.17,no.3,pp ,21. [11] R. O. Hryniv and Y. V. Mykytyuk, Half-inverse spectral problems for Sturm-Liouville operators with singular potentials, Inverse Problems,vol.2,no.5,pp ,24. [12] H. Koyunbakan and E. S. Panakhov, Half-inverse problem for diffusion operators on the finite interval, Mathematical Analysis and Applications,vol.326,no.2,pp ,27. [13] C.-F. Yang and Z.-Y. Huang, A half-inverse problem with eigenparameter dependent boundary conditions, Numerical Functional Analysis and Optimization,vol.31,no.4 6,pp , 21. [14] B. M. Levitan and I. S. Sargsjan, Sturm-Liouville and Dirac Operators, Kluwer Academic Publishers, Dordrecht, The Netherlands, [15]M.G.GasymovandB.M.Levitan, Theinverseproblemfor the Dirac system, Doklady Akademii Nauk SSSR, vol. 167, pp , [16] T. N. Arutyunyan, Isospectral Dirac operators, Izvestiya Akademii Nauk Armenii. Matematika,vol.29,no.2,pp.3 14,1994. [17] G. Borg, Eine umkehrung der sturm-liouvilleschen eigenwertaufgabe, bestimmung der Differential gleichung durch die eigenvert, Acta Mathematica,vol.78,pp.1 96,1946. [18] R. del Rio and B. Grébert, Inverse spectral results for AKNS systems with partial information on the potentials, Mathematical Physics, Analysis and Geometry, vol.4,no.3,pp , 21. [19] K. Mochizuki and I. Trooshin, Inverse problem for interior spectral data of the Sturm-Liouville operator, Inverse and Ill-Posed Problems,vol.9,no.4,pp ,21. [2] K. Mochizuki and I. Trooshin, Inverse problem for interior spectral data of the Dirac operator on a finite interval, Kyoto University. Research Institute for Mathematical Sciences,vol.38, no. 2, pp , 22. [21] C.-F. Yang and X.-P. Yang, An interior inverse problem for the Sturm-Liouville operator with discontinuous conditions, Applied Mathematics Letters,vol.22,no.9,pp ,29. [22] A. S. Ozkan and R. Kh. Amirov, An interior inverse problem for the impulsive Dirac operator, Tamkang Mathematics, vol. 42, no. 3, pp , 211. [23] R. Kh. Amirov, On a system of Dirac differential equations with discontinuity conditions inside an interval, Ukrainian Mathematical Journal,vol.57,no.5,pp ,25. [24] V. F. Zhdanovich, Formulas for the zeros of Dirichlet polynomials and quasipolynomials, Doklady Akademii Nauk SSSR, vol.13,no.5,pp ,196.

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