INVERSE PROBLEMS FOR STURM-LIOUVILLE OPERATORS WITH BOUNDARY CONDITIONS DEPENDING ON A SPECTRAL PARAMETER

Transcription

1 Electronic Journal of Differential Equations, Vol. 217 (217), No. 26, pp ISSN: URL: or INVERSE PROBLEMS FOR STURM-LIOUVILLE OPERATORS WITH BOUNDARY CONDITIONS DEPENDING ON A SPECTRAL PARAMETER MURAT SAT Communicated by Ira Herbst Abstract. In this article, we study the inverse problem for Sturm-Liouville operators with boundary conditions dependent on the spectral parameter. We show that the potential q(x) and coefficient a 1+b 1 functions can be uniquely c 1 +d 1 determined from the particular set of eigenvalues. 1. Introduction The theory of inverse problem for differential operators takes an important position in the trend development of the spectral theory of linear operators. Inverse problems of spectral analysis consist in recovering operators from their spectral characteristics 1, 7, 14, 16, 18, 21. Such problems often come along in mathematical physics, mechanics, electronics, geophysics and other branches of natural sciences. The inverse problem of a regular Sturm-Liouville operator was studied firstly by Ambarzumyan in and secondly by Borg in From then on, Borg s result has been extended to various versions. McLaughlin and Rundell in , established a new uniqueness theorem for the inverse Sturm-Liouville problem. They showed that the measurement of a particular set of eigenvalues was sufficient to define the obscure potential functions. They considered the eigenvalue problem y + ( q(x))y = y, < x < 1, y(, ) =, y (π, ) + H k y(π, ) =. They indicated that the spectral knowledge, for a constant index n (n =, 1, 2,... ), { n (q, H k )} + is equivalent to two spectra of boundary value problems with the equation and the first initial situation (one common boundary situation at x = ) and the second boundary situation (two different boundary conditions at x = π). In 19 the spectral data was handled by the Hochstadt and Lieberman method 17. Wang 22, 23 discussed the inverse problem for uncertain Sturm-Liouville operators on the finite interval a, b and diffusion operators. Here, we consider inverse spectral problems for Sturm Liouville operators with boundary conditions 21 Mathematics Subject Classification. 34B5, 34L2, 47E5. Key words and phrases. Inverse problem; uniqueness theorem; eigenvalue; spectral parameter. c 217 Texas State University. Submitted May 5, 216. Published January 24,

2 2 M. SAT EJDE-217/26 dependent on the spectral parameter with the above spectral knowledge. As far as we know, inverse spectral problems for Sturm Liouville operators with boundary conditions depending on the spectral parameter have not been studied with the spectral data before. Eigenvalue dependent boundary conditions have been studied extensively. References 3, 4, 5, 6, 8, 1, 12, 13 are well known examples for problems with boundary conditions depending linearly on the eigenvalue parameter. Recently inverse problems according to various spectral knowledge for eigenparameter linearly dependent Sturm-Liouville operator have been studied in 9, 11, 15, 2, 24, 25, 26, 27. We consider the Sturm-Liouville operator L := L(q, H k ) defined by Ly y + q(x)y = y, x π, (1.1) with boundary conditions dependent on the spectral parameter (a 1 + b 1 )y(, ) (c 1 + d 1 )y (, ) =, (1.2) y (π, ) + H k y(π, ) =. (1.3) Also we consider the Sturm-Liouville operator L := L( q, H k ) defined by Ly ỹ + q(x)ỹ = ỹ, ( x π), (1.4) with boundary conditions depending on the spectral parameter (ã 1 + b 1 )ỹ(, ) ( c 1 + d 1 )ỹ (, ) =, (1.5) ỹ (π, ) + H k ỹ(π, ) =, (1.6) where a 1, b 1, c 1, d 1, ã 1, b 1, c 1, d 1, H k R, such that δ 1 = a 1 d 1 b 1 c 1 <, δ 1 = ã 1 d1 b 1 c 1 <, < H 1 < H 2 < < H k < H k+1 < < H, the potentials q(x) and q(x) are real valued functions, q(x), q(x) L 1, π and is a spectral parameter. For the boundary-value problem (1.1)-(1.3) with coefficient H = (a2+b2) (c where 2+d 2) c 2 and d 2, instead of H k describes the actual background of Sturm Liouville operators with boundary conditions dependent on a spectral parameter; see 12. In this article, we construct a uniqueness theorem for Sturm-Liouville operators with boundary conditions depending on the spectral parameter on the finite interval, π. i.e. for a constant index n N, we demonstrate that if the spectral set { n (q, H k )} + for different H k can be restrained, then the spectral set { n (q, H k )} + a1+b1 is sufficient to define the potential q(x) and coefficient c 1+d 1 of the boundary condition. The techniques used here will be adopted from 17, 19, 26. Lemma 1.1 (4, 26). Eigenvalues n (n ) of the boundary-value problem (1.1)- (1.3) for coefficient H = H k = (a2+b2) (c 2+d 2) in (1.3) are roots of (1.3) and satisfy the asymptotic formula n = n O( 1 ). (1.7) n Lemma 1.2 (26). The solution to the (1.1) with the initial conditions y(, ) = (c 1 + d 1 ) and y (, ) = (a 1 + b 1 ) is y(x, ) = (c 1 + d 1 ) cos x + A(x, t) cos x + (a 1 + b 1 ) sin + 1 B(x, t) sin (1.8)

3 EJDE-217/26 INVERSE PROBLEMS FOR STURM-LIOUVILLE OPERATORS 3 where the kernel A(x, t) satisfies where q(x) = 2 da(x,x) dx 2 A(x, t) x 2 q(x)a(x, t) = 2 A(x, t) t 2,, A(, ) = h, A(x,t) t where q(x) = 2 db(x,x) dx, B(x, ) =. 2 B(x, t) x 2 q(x)b(x, t) = 2 B(x, t) t 2 2. Main results and their proofs t= = ; and the kernel B(x, t) satisfies Theorem 2.1. Let σ(l kj ) := { n (q, H kj )} (j = 1, 2) be the spectrum of the boundary value problem (1.1)-(1.3) with coefficient H kj. If H k1 H k2, then where k j N, and denotes an empty set. σ(l k1 ) σ(l k2 ) = (2.1) Lemma 2.2. Let n (q, H k ) be the n-th eigenvalue of the boundary-value problem (1.1)-(1.3). Then the spectral set { n (q, H k )} + is a bounded infinite set, where < H 1 < H 2 < < H k < H k+1 < < H. The above Lemma carries a significant part in the proof of the next theorem. Theorem 2.3. Let n (q, H k ) be the n-th eigenvalue of the boundary-value problem (1.1)-(1.3) and n ( q, H k ) be the n-th eigenvalue of the boundary-value problem (1.4)- (1.6), for a constant index n(n N). If n (q, H k ) = n ( q, H k ) for all k N, then q(x) = q(x)a.e. on, π, ã 1 + b 1 c 1 + d 1 = a 1 + b 1 c 1 + d 1, C. Proof of Theorem 2.1. Suppose the argument of Theorem 2.1 is false. Then there exists n1 (H k1 ) = n2 (H k2 ) R, where nj (H kj ) σ(l kj ) for j = 1, 2 and n j N. Let y kj (x, nj (H kj )) be the solution of (1.1)-(1.3) with the eigenvalue nj (H kj ) and satisfy the initial conditions y kj (, nj (H kj )) = (c 1 + d 1 ) and y k j (, nj (H kj )) = (a 1 + b 1 ). For a fixed index n, we have and y k 1 (x, n1 (H k1 )) + q(x)y k1 (x, n1 (H k1 )) = n1 (H k1 )y k1 (x, n1 (H k1 )) (2.2) y k 2 (x, n2 (H k2 )) + q(x)y k2 (x, n2 (H k2 )) = n2 (H k2 )y k2 (x, n2 (H k2 )). (2.3) By multiplying (2.2) by y k2 (x, n2 (H k2 )) and (2.3) by y k1 (x, n1 (H k1 )) respectively and subtracting and integrating from to π, we obtain Using the initial conditions, we obtain (y k2 y k 1 y k1 y k 2 ) π =. (2.4) y k2 (π, n2 (H k2 ))y k 1 (π, n1 (H k1 )) y k1 (π, n1 (H k1 ))y k 2 (π, n2 (H k2 )) =. (2.5)

4 4 M. SAT EJDE-217/26 On the other hand, we have the equality y k2 (π, n2 (H k2 ))y k 1 (π, n1 (H k1 )) y k1 (π, n1 (H k1 ))y k 2 (π, n2 (H k2 )) = y k2 (π, n2 (H k2 ))y k 1 (π, n1 (H k1 )) + H k1 y k1 (π, n1 (H k1 )) y k1 (π, n1 (H k1 ))y k 2 (π, n2 (H k2 )) + H k2 y k2 (π, n2 (H k2 )) + (H k2 H k1 )y k1 (π, n1 (H k1 ))y k2 (π, n2 (H k2 )) = (H k2 H k1 )y k1 (π, n1 (H k1 ))y k2 (π, n2 (H k2 )). Since H k2 H k1, if y k1 (π, n1 (H k1 ))y k2 (π, n2 (H k2 )) =, it follows that (2.6) y k1 (π, n1 (H k1 )) = or y k2 (π, n2 (H k2 )). (2.7) By virtue of (2.7) together with (1.3), this yields or By (2.8) and (2.9), this yields y k1 (π, n1 (H k1 )) = y k 1 (π, n1 (H k1 )) = (2.8) y k2 (π, n2 (H k2 )) = y k 2 (π, n2 (H k2 )) =. (2.9) y k1 (x, n1 (H k1 )) = or y k2 (x, n2 (H k2 )) = on, π, (2.1) This is impossible. Thus, we obtain y k2 (π, n2 (H k2 ))y k 1 (π, n1 (H k1 )) y k1 (π, n1 (H k1 ))y k 2 (π, n2 (H k2 )). (2.11) Clearly, this contradicts (2.5); therefore (2.1)) holds. The proof is complete. Proof of Lemma 2.2. We will show that the following formula holds n (H ) < < n (H k+1 ) < n (H k ) < < n (H 1 ). (2.12) Let y(x, n (H)) be the solution of the boundary value problem (1.1)-(1.3) of the eigenvalue n (H) and satisfies the initial conditions y(, n (H)) = (c 1 + d 1 ) and y (, n (H)) = (a 1 + b 1 ). We have y (x, n (H)) + q(x)y(x, n (H)) = n (H)y(x, n (H)), (2.13) y (x, n (H + H)) + q(x)y(x, n (H + H)) = n (H + H)y(x, n (H + H)) (2.14) where H is the enhancement of H. Multiplying (2.13) by y(x, n (H + H)) and multiplying (2.14) by y(x, n (H)) and subtracting from each other and integrating from to π, we obtain n (H) y(x, n (H))y(x, n (H + H))dx = Hy(π, n (H)y(π, n (H + H)), (2.15) where n (H) = n (H + H) n (H). It is well understood that y(x, n (H)) and n (H) are real and continuous with respect to H. Dividing (2.15) by H, and letting H in (2.15), we have n (H) H y 2 (x, n (H))dx = y 2 (π, n (H)). (2.16) If y(π, n (H)) =, then y (π, n (H)) =. By the uniqueness theorem, this yields y(x, n (H)).

5 EJDE-217/26 INVERSE PROBLEMS FOR STURM-LIOUVILLE OPERATORS 5 This contradicts the eigenfunction y(x, n (H) corresponding to eigenvalue n (H). Hence y 2 (π, n (H)) > and y2 (x, n (H))dx >. From (2.16), we have n (H) H >. This implies that (2.12) holds. Therefore the spectral set { n (q, H k )} + is a bounded infinite set. The proof is complete. Finally, using Theorem 2.1, Lemma 2.2 and the properties of entire functions, we show that Theorem 2.3 holds. Proof of Theorem 2.3. According to Lemma 1.2, solutions to equation (1.1) with boundary condition (1.2) and the equation (1.4) with boundary condition (1.5) can be stated in the integral forms: y(x, ) = (c 1 + d 1 ) cos x + A(x, t) cos and + (a 1 + b 1 ) sin ỹ(x, ) = ( c 1 + d 1 ) cos x + + (ã 1 + b 1 ) sin yỹ = (c 1s 2 + d 1 )( c 1 s 2 + d 1 ) 2 x + 1 x + 1 B(x, t) sin (2.17) Ã(x, t) cos B(x, t) sin (2.18) respectively. Let = s 2. From (2.17), (2.18) and 26, proof of Theorem 2.1 we obtain 1 + cos 2sx + k(x, τ) cos 2sτdτ + (a 1s 2 + b 1 )(ã 1 s 2 + b 1 ) 2s 2 1 cos 2sx s (c 1s 2 + d 1 )(ã 1 s 2 + b 1 ) sin 2sx s ( c 1s 2 + d 1 )(a 1 s 2 + b 1 ) sin 2sx + h(x, τ) cos 2sτdτ l(x, τ) sin 2sτdτ m(x, τ) sin 2sτdτ, (2.19) where the functions k(x, τ), h(x, τ), l(x, τ) and m(x, τ) are continuous functions. We define the function w() = (a 2 + b 2 )y(π, ) (c 2 + d 2 )y (π, ). From (2.17), we obtain the asymptotic forms Hence y(π, ) = (c 1 + d 1 ) cos π + O( e Im π ), y (π, ) = (c 1 + d 1 ) sin π + O( e Im π ). w() = (c 1 + d 1 )(c 2 + d 2 ) sin π + O( 2 e Im π ). (2.2)

6 6 M. SAT EJDE-217/26 Zeros of w() are the eigenvalues of the Sturm-Liouville problem (1.1)-(1.3) where H k = H = (a2+b2) (c. w() is an entire function of order 1 2+d 2) 2 of. Multiplying (1.4) by y, (1.1) by ỹ and subtracting and integrating from to π, we take (ỹy yỹ ) π + ( q q)yỹdx =. Using y(, ) = (c 1 + d 1 ), ỹ(, ) = ( c 1 + d 1 ), y (, ) = (a 1 + b 1 ) and ỹ (, ) = (ã 1 + b 1 ), this yields = ỹ(π, )y (π, ) y(π, )ỹ (π, ) + (a 1 + b 1 )( c 1 + d 1 ) (c 1 + d 1 )(ã 1 + b 1 ) + Let Q(x) = ( q(x) q(x)) and ( q(x) q(x))yỹdx. K() = (a 1 c 1 ã 1 c 1 ) 2 + (a 1 d1 + b 1 c 1 ã 1 d 1 b 1 c 1 ) + (b 1 d1 b 1 d 1 ) + Q(x)yỹdx. (2.21) (2.22) Clearly, the function K() is an entire function. Because the first term of equation (2.21) for = n (q, H k ) is zero, then K( n (q, H k )) =. From Lemmas 1.1 and 2.2, we see that the spectral set { n (q, H k )} + is a bounded infinite set. Therefore, if consists of n (q) R, such that n (q) is a finite accumulation dot of the spectrum set { n (q, H k )} +. It is well understood that the set of zeros of every entire function which is not identically zero hasn t any finite accumulation dot. Hence K() =, C. (2.23) From (2.22), (2.23) and 26, proof of Theorem 2.1, we have Q(x) = q(x) q(x) =, a.e. on, π, The proof is complete. ã 1 + b 1 c 1 + d 1 = a 1 + b 1 c 1 + d 1, C. References 1 D. Alpay, I. Gohberg; Inverse problems associated to a canonical differential system, Recent Advances in Operator Theory and Related Topics. Birkhäuser Basel, 127 (21), V. A. Ambartsumyan; Über eine frage der eigenwerttheorie, Zeitschrift für Physik, 53 (1929), W. F. Bauer; Modified Sturm-Liouville systems, Quarterly of Applied Mathematics, 11 (1953), P. A. Binding, P. J. Browne, K. Seddighi; Sturm-Liouville problems with eigenparameter dependent boundary conditions, Proceedings of the Edinburgh Mathematical Society (Series 2), 37 (1994), P. A. Binding, P. J. Browne, B. A. Watson; Inverse spectral problems for Sturm Liouville equations with eigenparameter dependent boundary conditions, Journal of the London Mathematical Society, 62 (2),

7 EJDE-217/26 INVERSE PROBLEMS FOR STURM-LIOUVILLE OPERATORS 7 6 P. A. Binding, P. J. Browne, B. A. Watson; Recovery of the m-function from spectral data for generalized Sturm-Liouville problems, Journal of computational and applied mathematics, 171 (24), G. Borg; Eine umkehrung der Sturm-Liouvillesehen eigenwertaufgabe, 78 (1945), P. J. Browne, B. D. Sleeman; Inverse nodal problems for Sturm-Liouville equations with eigenparameter dependent boundary conditions, Inverse Problems, 12 (1996), S. Buterin; On half inverse problem for differential pencils with the spectral parameter in boundary conditions, Tamkang journal of mathematics, 42 (211), W. Feller; The parabolic differential equations and the associated semi-groups of transformations, Annals of Mathematics, 55 (1952), G. Freiling, V. A. Yurko; Inverse problems for Sturm-Liouville equations with boundary conditions polynomially dependent on the spectral parameter, Inverse Problems 26 (21), C. T. Fulton; Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 77A (1977), R. E. Gaskell; A problem in heat conduction and an expansion theorem, American Journal of Mathematics, 64 (1942), F. Gesztesy, B. Simon; A new approach to inverse spectral theory. II: General real potentials and the connection to the spectral measure, Annals of Mathematics, 152 (2), N. J. Guliyev; The regularized trace formula for the Sturm-Liouville equation with spectral parameter in the boundary conditions, Proc. Inst. Math. Natl. Alad. Sci. Azerb., 22 (25), H. Hochstadt; The inverse Sturm Liouville problem, Communications on Pure and Applied Mathematics, 26 (1973), H. Hochstadt, B. Lieberman; An inverse Sturm-Liouville problem with mixed given data, SIAM Journal on Applied Mathematics, 34 (1978), B. M. Levitan; On the determination of the Sturm-Liouville operator from one and two spectra, Izvestiya: Mathematics, 12 (1978), J. R. McLaughlin, W. Rundell; A uniqueness theorem for an inverse Sturm-Liouville problem, Journal of Mathematical Physics, 28 (7) (1987), A. S. Ozkan, B. Keskin; Inverse nodal problems for Sturm-Liouville equation with eigenparameter-dependent boundary and jump conditions, Inverse Problems in Science and Engineering, 23 (215), J. Poschel, E. Trubowitz; Inverse Spectral Theory, Academic Press Orlando, 13, Y. P. Wang; A uniqueness theorem for indefinite Sturm-Liouville operators, Appl. Math. J. Chinese Univ., 27 (3) (212), Y. P. Wang; A uniqueness theorem for diffusion operators on the finite interval, Acta Mathematica Scienta, 33(2) (213), Y. P. Wang; Inverse problems for Sturm Liouville operators with interior discontinuities and boundary conditions dependent on the spectral parameter, Mathematical Methods in the Applied Sciences, 36 (213), Y. P. Wang, C. T. Shieh; Inverse Problems for Sturm-Liouville Equations with Boundary Conditions Linearly Dependent on the Spectral Parameter from Partial Information, Results in Mathematics, 65 (214), Y. P. Wang, C. F. Yang, Z. Y. Huang; Half inverse problem for Sturm-Liouville operators with boundary conditions dependent on the spectral parameter, Turkish Journal of Mathematics, 37 (213), C. F. Yang; New trace formula for the matrix Sturm-Liouville equation with eigenparameter dependent boundary conditions, Turkish Journal of Mathematics, 37 (213), Murat Sat Department of Mathematics, Faculty of Science and Art, Erzincan University, Erzincan, 241, Turkey address: murat sat24@hotmail.com