AN INVERSE SPECTRAL PROBLEM FOR STURM-LIOUVILLE OPERATOR WITH INTEGRAL DELAY

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1 Electronic Journl of Differentil Equtions, Vol. 217 (217), No. 12, pp ISSN: URL: or AN INVERSE SPECTRAL PROBLEM FOR STURM-LIOUVILLE OPERATOR WITH INTEGRAL DELAY MANAF DZH. MANAFOV In memory of M. G. Gsymov ( ) Abstrct. In this rticle, we study n inverse spectrl problem for Sturm- Liouville opertor with integrl dely. We prove tht the stndrd spectrl symptotic conditions re necessry nd sufficient for unique solvbility of the inverse problem. 1. Introduction We consider inverse problem for the boundry-vlue problem (BVP) generted by the integro-differentil eqution l y := y + q(x)y + with the Dirichlet boundry conditions M(x t)y(t)dt = 2 y, x (, ) (, π) (1.1) U(y) := y() =, V (y) := y(π) =, (1.2) nd the conditions t the point x = : y( + ) = y( ) y(), I(y) := y ( + ) y ( ) = 2αy(), (1.3) q(x) nd M(x) re complex-vlued functions, q(x) L 2 (, π) nd (π x)m(x) L 2 (, π), α C, ( π 2, π) nd is spectrl prmeter. Sturm-Liouville spectrl problems with potentils depending on the spectrl prmeter (in cse K(x) ) rise in vrious models of quntum nd clssicl mechnics. For instnce, the evolution equtions tht re used to model interctions between colliding reltivistic spineless prticles cn be reduced to the form (1.1). Then 2 is ssocited with the energy of the system (see [12, 13]). Spectrl problems of differentil opertors re studied in two min brnches, nmely, direct nd inverse problems. Direct problems of spectrl nlysis consist in investigting the spectrl properties of n opertor. On the other hnd, inverse problems im t recovering opertors from their spectrl chrcteristics. Such problems often pper in mthemtics, mechnics, physics, electronics, geophysics, meteorology nd other brnches of nturel sciences nd engineering. Direct nd 21 Mthemtics Subject Clssifiction. 34A55, 34L5, 47G2. Key words nd phrses. Sturm-Liouville Opertor; inverse spectrl problem; integrl dely. c 217 Texs Stte University. Submitted October 2, 216. Published Jnury 12,

2 2 MANAF DZH. MANAFOV EJDE-217/12 inverse problems for the clssicl Sturm-Liouville opertors hve been extensively studied (see [5, 7, 11] nd the references therein). For integro-differentil nd other clsses of nonlocl opertors inverse problems re more difficult for investigtion, nd the clssicl methods either re not pplicble to them or require essentil modifictions (see [1, 2, 3, 5, 6, 14, 15]). In this spect, vrious inverse spectrl problems for the (1.1), (1.3) BVP (specil cse M(x) ) hve been investigted in [8, 9, 1]). In this rticle we estblish uniqueness result for inverse spectrl problem for Sturm-Liouville opertor with integrl dely. 2. Integrl representtions for solutions In this section, we construct n integrl representtion of the solution y(x, ) of (1.1), (1.3), stisfying the initil conditions y(, ) = 1, y (, ) = i. (2.1) Also we study some properties of the solutions. Using the stndrd successive pproximtion methods (see [11]), we cn prove the following theorem. Theorem 2.1. The solution y(x, ) hs the form where y(x, ) = y (x, ) + y (x, ) = nd the function A(x, t) stisfies with nd C = α. σ (x) = e ix, x < (1 iα)e ix + iαe i, x > A(x, t)e it dt, (2.2) A(x, t) dt e Cσ(x) 1 (2.3) (x t)[ q(t) + t M(t τ) dτ]dt, Proof. It is cler tht when α =, if we consider the eqution (1.1) seprtely on the intervls (, ) nd (, π), we cn write the solutions s e (x, ) = e ix + e (x, ) = e i(x ) + K (x, t)e it dt, x <, (2.4) +2 K (x, t)e i(t ) dt, x >, (2.5) respectively. For the solutions of the bove equtions to solve the eqution tht hs representtion (2.5), the following equlity must be stisfied: +2 = 1 + t K (x, t)e i(t ) dt [ t sin (x t) q(t) e i(t ) + t+2 ] K (t, τ)e i(τ ) dτ [ τ M(t τ) e i(τ ) + K (τ, s)e i(s ) ds τ+2 ] dτ } dt.

3 EJDE-217/12 AN INVERSE SPECTRAL PROBLEM 3 It is esy to obtin the integrl eqution K (x, t) = t 2 q(u)du t+(x u) t (x u) u t+(x u) t (x u) q(u) t+(x u) t (x u) M(u v) dx du K (u, v) dv du M(u v)k (v, ξ)dξ dx du. (2.6) Since e (x, ) is lso the solution of (1.1), (1.3) on the intervl < x π, the solution y(x, ) hs the form e (x, ), x <, y(x, ) = (2.7) c 1 e (x, ) + c 2 e (x, ), < x π, where the constnts c 1, c 2 re defined from conditions (1.3). Hence, we hve e (x, ), x <, y(x, ) = e (, ) (1 2iα)e(x,)+(1+2iα)e(x, ) (2.8) 2 +e (, ) e(x,) e(x, ) 2i, < x π. Using (2.4), (2.5) nd (2.8), fter some simple computtions, we find the following expression for y(x, ) ( < x π), where y(x, ) = e(x, ) + +2 e(x, ) = e (, )[cos (x ) + 2α sin (x )] + e (, ) = (1 iα)e ix + iαe i(2) + K (x, t)e(t, )dt, (2.9) A 1 (x, t)e it dt, t+x sin (x ) (2.1) A 1 (x, t) = A K (, t + 2 x) K (, t + x) + 1 H(s)ds, t < x, 2 t+2 1 A = 2 q(t)dt + 1 +t 4 q( 2 )dt, 2 x < t < x,, < t < 2 x, H(s) = 1 2 s 2 K (σ, s + σ)q(σ)dσ s 2 K (σ, s + σ)q(σ)dσ. (2.11) Here, we ssume tht K (, t), H(t), for t > nd A 1 (x, t) = for t > x. Now using the expression (2.1) in (2.9), we hve for < x π ( t < x) where y(x, ) = (1 iα)e ix + iαe i(2) + A 2 (x, t)e it dt, (2.12) A 2 (x, t) = A 1 (x, t) + (1 iα)k (x, t) iαk (x, 2 t) + 2 K (x, s)a 1 (s, t)ds. (2.13)

4 4 MANAF DZH. MANAFOV EJDE-217/12 From (2.4) nd (2.12), we cn write the formul (2.2) for the solution y(x, ), where K (x, t), if x, t < x A(x, t) = (2.14) A 2 (x, t), if < x π, t < x. From (2.6) it is esy to obtin 2 where C > is constnt nd σ (x) = K (x, t) dt e Cσ(x) 1, (2.15) (x t) [ q(t) + t M(t τ) dτ ] dt. Using (2.15), from (2.11) nd (2.13), we hve the estimte A 2 (x, t) dt e Cσ(x) 1 (2.16) for some constnt C >. Hence, from (2.14) nd (2.16), we rrive t (2.3). Let s(x, ) be solution of (1.1) with initil conditions s(, ) =, s (, ) = 1. Becuse y(x, ) nd y(x, ) re two linerly independent solutions of (1.1), (1.3), then y(x, ) y(x, ) s(x, ) =. 2i Using integrl representtion (2.2), we esily obtin s(x, ) = s (x, ) + G(x, t) sin t dt, (2.17) where sin x s (x, ) = x < sin x sin (2) (1 iα) + iα, x >, G(x, t) = A(x, t) A(x, t) is continuous function, nd G(x, ) =. 3. Properties of the spectrl chrcteristics In the section, we study properties of eigenvlues nd eigenfunctions of (1.1). Let y(x) nd z(x) be continuously differentible functions on (, ) nd (, π). Denote y, z := yz y z. If y(x) nd z(x) stisfy the mtching conditions (1.3), then y, z x= = y, z x=+, (3.1) i.e. the function y, z is continuous on (, π). Denote () = s(π, ). The eigenvlues 2 n} n 1 of the BVP (1.1) coincide with the zeros of the function (). Theorem 3.1. The eigenvlues 2 n nd eigenfunctions s(x, n ) of the BVP (1.1) stisfy the following symptotic estimtes for sufficiently lrge n, n = n + o ( 1 ), (3.2) n

5 EJDE-217/12 AN INVERSE SPECTRAL PROBLEM 5 s(x, n ) = o ( 1 ) + (1 iα) sin n x sin n x, x < n n + iα sin n (2), x >, n (3.3) where sin π sin (2 π) n re the roots of () := (1 iα) + iα nd n = n + h n, h n l. Proof. From (2.17), we hve sin π sin (2 π) () = (1 iα) + iα + G(π, t) sin t dt. (3.4) Denote Γ n := : = n + δ}, n =, 1,..., (δ > ). Since () () = Im π e e o( ) nd () C Im π δ for ll Γ n, we estblish by the Rouche s Theorem (see [4, p. 125]) tht n = n+ε n, where ε n = o(1). Moreover, ε n = o( 1 ) n is obtined from the equlity o = ( n ) = ( ( n) + o(1))ε n + o( 1 n ). This completes the proof of (3.2). From (2.17) nd (3.2), one cn esily prove tht the symptotic formul (3.3) is true. Theorem 3.2. The specifiction of the spectrum 2 n} n 1 uniquely determines the chrcteristic function () by the formul 2 n 2 () = [(1 iα)π + iα(2 π)] ( n) 2. (3.5) Proof. It follows from (3.4) nd consequently by Hdmrd s fctoriztion theorem (see [4, p. 289]), () is uniquely determined up to multiplictive constnt by its zeros: () = C (1 2 ). (3.6) Consider the function Then () := (1 iα) sin π = [(1 iα)π + iα(2 π)] () () = C 1 [(1 iα)π + iα(2 π)] 2 n sin (2 π) + iα (1 2 ( n) 2 ). ( n) 2 2 (1 + 2 n ( n) 2 ) ( n) 2 2. Tking (3.2) nd (3.4) into ccount we clculte () lim (1 () = 1, lim + 2 n ( n) 2 ) ( n) 2 2 = 1 nd hence C = [(1 iα)π + iα(2 π)] Substituting this into ccount (3.6) we rrive t (3.5). 2 n ( n) 2.

6 6 MANAF DZH. MANAFOV EJDE-217/12 4. Formultion of the inverse problem uniqueness theorem In this section, we study inverse problem of recovering M(x) from the given spectrl chrcteristics. We denote the BVP (1.1)-(1.3) by L = L(M). Together with L = L(M) we consider BVP L = L( M) of the sme form, but with different kernel M. Inverse Problem: Given function q(x), numbers α,, nd the spectrum n } n 1, construct the function M(x). Let us prove the uniqueness theorem for the solution of the Inverse Problem. Everywhere below if certin symbol e denotes n object to L, then the corresponding symbol ẽ denotes the nlogous object relted to L nd ê = e ẽ. Theorem 4.1. Fix b (, ). Let Λ N be subset of nonnegtive integer numbers, nd let Ω := 2 n} n Λ be prt of the spectrum of L such tht the system of functions cos n x} n Λ is complete in L 2 (, π). Let M(x) = M(x) lmost everywhere (.e.) on (b, π), nd Ω = Ω. Then M(x) = M(x).e. on (, π). Proof. Let χ(x, ) be the solution of the eqution l z := z + q(x)z + x M(t x)z(t)dt = 2 z, x (, ) (, π) (4.1) under the conditions χ(π, ) =, χ (π, ) = 1 nd the conditions t the point x = : χ( +, ) = χ(, ) χ(, ), χ ( +, ) χ (, ) = 2αχ(, ). Denote () = χ(, ). Then by (3.1) we hve = = χ(x, ) M(x t) s(t, ) dt dx χ(x, )l s(x, )dx l χ(x, ) s(x, )dx χ(x, ) l s(x, )dx χ(x, ) l s(x, )dx + [ s(x, )χ (x, ) s (x, )χ(x, )]( + π ) = () (). For l = l we hve () (), nd consequently χ(x, ) We trnsform (4.2) into ( M(x) Denote w(x, ) = χ(π x, ), N(x) = M(π x), Then (4.2) tkes the form x ϕ(x, ) = M(x t) s(t, ) dt dx = (). (4.2) ) χ(t, ) s(t x, )dt dx = (). (4.3) w(t, ) s(x t, )dt. (4.4) N(x)ϕ(x, )dx = (). (4.5)

7 EJDE-217/12 AN INVERSE SPECTRAL PROBLEM 7 For x the following representtion holds [14], ϕ(x, ) = 1 ( ) 2 2 x cos x + V (x, t) cos t dt, (4.6) where V (x, t) is continuous function which does not depend on. Since Ω = Ω, we hve by Theorem 3.2 () () = (). Then, substituting (4.6) into (4.5), we obtin b ( b x N(x) + V (t, x) N(t) ) cos x dx, nd consequently, N(x) + b x x V (t, x) N(t)dt =.e. on (, b). Since this homogeneous Volterr integrl eqution hs only the trivil solution it follows tht N(x) =.e. on (, b), i.e. M(x) = M(x).e. on (, π). Acknowledgements. The uthor wnts to thnk the nonymous referees for their vluble suggestions tht improving this rticle. This work ws supported by Grnt No FEFMAP/216-2 from Adiymn University of Reserch Project Coordintion (ADYUBAP), Turkey. References [1] Buterin, S. A.; On n inverse spectrl problem for convolution integro-differentil opertor, Result. Mth., 5 (27), [2] Buterin, S. A.; On the reconstruction of convolution perturbtion of the Sturm-Liouville opertor from the spectrum, Diff. Urvneniy, 46:1 (21), ; English trnsl.: Diff. Equtions, 46:1 (21), [3] Buterin, S. A.; Choque Rivero, A. E.; On inverse problem for convolution integro-differentil opertor with Robin boundry conditions, Appl. Mth. Letters, 48 (215), [4] Conwy, J. B.; Functions of One Complex Vrible. Springer-Verlg, New York, USA 2nd ed., [5] Frelling, G.; Yurko, V.; Inverse Sturm-Liouville Problems nd Their Applictions, Nov Science Publ., Inc: Huntington, NY, 21. [6] Kuryshov, Ju. V.; Inverse spectrl problem for integro-differentil opertors, Mth. Zmetki, 81:6 (27), ; English trnsl. Mth. Notes, 81:6 (27), [7] Levitn, B. M.; Inverse Sturm-Liouville Problems. VSP:Zeist, [8] Mnfov, M. Dzh.; Hlf-inverse spectrl problem for differentil pensils with interction-point nd eigenvlue-dependent boundry conditions. Hcettepe J. of Mth. nd Stts., 42:4 (213), [9] Mnfov, M. Dzh.; Kbln, A.; Inverse spectrl nd inverse nodl problems for energydependent Sturm-Liouville equtions with δ-interction, Electronic J. of Diff. Equtions, vol. 215 (215), no. 26, 1-1. [1] Mnfov, M. Dzh.; Inverse spectrl problems for energy-dependent Sturm-Liouville equtions with finitely mny point δ-interctions, Electronic J. of Diff. Equtions, vol. 216 (216), no. 11, [11] Mrchenko, V. A.; Sturm-Liouville Opertors nd Their Applictions. Opertor Theory: Advnced nd Appliction, Birkhuser: Bsel, 22, [12] Mrkus, A. S.; Introduction to the Spectrl Theory of Polynomil Opertor Pensils. Shtinits, Kishinev, 1986; English trnsl., AMS, Providense, [13] Jons, P.; On the spectrl theory of opertors ssocited with perturbed Klein-Gordon nd wve type equtions. J. Opertor Theory, 29 (1993),

8 8 MANAF DZH. MANAFOV EJDE-217/12 [14] Yurko, V. A.; An inverse problem for integro-differentil opertors, Mt. Zmetki, 5:5 (1991), ; English trnsl.: Mth. Notes, 5:5-6 (1991), [15] Yurko, V. A.; An inverse spectrl problems for integro-differentil opertors, Fr Est J. Mth. Sci. 92:2 (214), Mnf Dzh. Mnfov Adıymn University, Fculty of Science nd Arts, Deprtment of Mthemtics, 24, Adıymn, Turkey E-mil ddress: