THE MAIN EQUATION OF INVERSE PROBLEM FOR DIRAC OPERATORS

Transcription

1 U.P.B. Sci. Bull., Series A, Vol. 79, Iss. 4, 7 ISSN 3-77 THE MAIN EQUATION OF INVERSE PROBLEM FOR DIRAC OPERATORS Ozge Akcy, Khnlr R. Mmedov In this pper, the min eqution or Gelfnd-Levitn-Mrchenko type eqution of inverse problem for Dirc opertors with piecewise continuous coefficient is derived. The uniqueness theorem for inverse spectrl problem ccording to the sequences of eigenvlues nd normlized numbers is proved. Keywords: Dirc opertor, min eqution, inverse problem. MSC: 34A55, 34L4.. Introduction The theory of inverse problems for differentil opertors plys n importnt role in the development of the spectrl theory of liner opertors. The inverse spectrl problem is the reconstruction of liner opertor from some of its spectrl chrcteristics, such s spectrl dt, spectrl function, spectr for different boundry conditions, scttering dt, Weyl function, etc. According to the spectrl chrcteristic, different inverse problems cn be considered. The most comprehensive informtion on the theory of inverse problems cn be found in the books [4, 8, 3. The direct nd inverse problems for Dirc opertors hve ttrcted considerble ttention in both mthemtics nd physics. Especilly, since Dirc eqution is relted to nonliner wve eqution this ws discovered in [, 3, there hs been mny investigtions bsed on Dirc eqution nd the investigtions hve been continuing to be developed in mny directions. In this pper, our im is to prove the uniqueness theorem of inverse problems for Dirc opertors with discontinuous coefficient ccording to the sequences of eigenvlues nd normlized numbers nd give n lgorithm to construct the potentil function. Then, we consider the following boundry vlue problem B Bu Qxu λrxu, < x < π, u λ h u π h u π, q x q, Q x x q x q x u x, u u x the functions q x L, π nd q x L, π re rel vlued, λ is spectrl prmeter, {, x, rx α, < x π, < α, h nd h > re rel numbers. Mersin University, Science nd Letters Fculty, Deprtment of Mthemtics, Mersin, 33343, Turkey, e-mil: ozge.kcy@gmil.com Mersin University, Science nd Letters Fculty, Deprtment of Mthemtics, Mersin, 33343, Turkey, 59

2 6 Ozge Akcy, Khnlr R. Mmedov In the finite intervl, in the cse of rx in the eqution. nd the potentil function Qx is continuous, the solvbility of inverse problem ccording to two spectr ws exmined in [5 nd ccording to one spectrum nd normlized numbers ws given in [. The inverse problem contined spectrl prmeter in boundry condition by spectrl function ws studied in [9. Inverse spectrl problems for Dirc opertor with summble potentil were worked in [4, 7. An lgorithm for reconstructing the Dirc opertor ws given in [,, 5,,. The uniqueness theorem of the inverse problem for Dirc opertors with spectrl prmeter in boundry conditions by Weyl function ws proved in [. Moreover, the works [6, 8, 9 cn be exmined for the physicl pplictions of Dirc eqution. As different from other works, the problem.,. hs piecewise continuous coefficient, so the integrl representtion not opertor trnsformtion for the solution of eqution. obtined in [6 is used. This pper is orgnized s follows: In section, we give n opertor formultion of the problem.,. nd the symptotic behviour of eigenvlues, eigenfunctions nd normlized numbers of problem.,. obtined by using the integrl representtion. In section 3, Gelfnd-Levitn-Mrchenko type eqution with respect to the kernel of this integrl representtion is derived nd it is obtined tht this eqution hs unique solution. Then, we prove the uniqueness theorem for the solution of inverse problem by its eigenvlues nd normlized numbers. Finlly, we give n lgorithm to construct the potentil function Qx.. Opertor Formultion nd Some Spectrl Properties We denote the inner product in Hilbert spce H r L,r, π; C C by U, V : [u xv x u xv x rxdx h u 3 v 3, U u x, u x, u 3 T H r, V v x, v x, v 3 T H r. Let us define the opertor L by LU : with the domin DU lu h u π h u π { U U u x, u x, u 3 T H r, u x, u x AC[, π, u 3 u π, lu L,r, π; C, u lu rx Bu Qxu. Thus, the considered problem.,. is equivlent the eqution LU λu. Denote by ϕx, λ nd ϑx, λ the solutions of the eqution. under the initil conditions ϕ, λ, ϕ, λ, 3 ϑ π, λ h, ϑ π, λ λ h. The integrl representtion of the solution ϕx, λ hs the form µx sin λy ϕx, λ ϕ x, λ Gx, y cos λy ϕ x, λ sin λµx cos λµx } dy, 4, µx { x, x, αx α, < x π,

3 The Min Eqution of Inverse Problem for Dirc Opertors 6 G ij x,. L, π, i, j, for fixed x [, π see [6,. Moreover, the kernel Gx, y stisfies the following problem BG xx, y rxg yx, yb QxGx, y, Qx rx[gx, µxb BGx, µx, 5 G x, G x,. Here, we specify tht the reltion.3 expresses the connection between the kernel Gx, t nd the potentil function Qx of the eqution. nd this reltion is used to prove the uniqueness theorem for inverse problem. Define the chrcteristic function ωλ of L by Then, it follows from.4 tht ωλ : ϕ x, λϑ x, λ ϕ x, λϑ x, λ. 6 ωλ ϑ, λ λ h ϕ π, λ h ϕ π, λ. Lemm.. [ The zeros λ n of the chrcteristic function ωλ coincide with the eigenvlues of the problem L. The functions ϕx, λ n nd ϑx, λ n re eigenfunctions nd there exists the sequence κ n such tht ϑx, λ n κ n ϕx, λ n, κ n. 7 The normlized numbers re defined by : ϕ x, λ n ϕ x, λ n rxdx ϕ π, λ n. h Lemm.. [ The reltion is vlid, ωλ d dλ ωλ. ωλ n κ n, 8 Remrk.. The following estimtes re obtined by using the integrl representtion. s λ uniformly in x [, π ϕ x, λ sin λµx O, find ϕ x, λ cos λµx O λ e Imλ µx λ e Imλ µx Let us substitute the estimtes.7 in the chrcteristic function ωλ. Then, we ωλ λ sin λµπ O e Imλ µπ, λ. Theorem.. The symptotic formuls for eigenvlues λ n for n Z, eigenfunctions nd normlized numbers of boundry vlue problem.,. re s follows, respectively: λ n λ n ϵ n, {ϵ n } l,. 9 ϕx, λ n sin nπµx µπ cos nπµx µπ ζ n x x ζ n, λ n nπ µπ, { ζ n } x l nd µπ τ n, { } {τ n } l, 3 x l for ll x [, π. ζ n Proof. The proof of this theorem is similrly obtined in [.

4 6 Ozge Akcy, Khnlr R. Mmedov Moreover, in considertion of.8, since the function ωλ is entire function, it is obtined from Hdmrd s theorem see [7 tht ωλ µπλ λ λ n λ λ n. 4 Lemm.3. [ The eigenfunction expnsion formul gx β n ϕx, λ n, β n gx, ϕx, λ n 5 holds for the bsolutely continuous function gx, x [, π nd the series converges uniformly in x [, π. n 3. The Uniqueness Theorem for Inverse Problem In this section, the uniqueness of the solution of inverse problem will be proved by using the Gelfnd-Levitn-Mrchenko method. In this method, the trnsformtion opertor is used nd the min role is plyed by liner integrl eqution with respect to the kernel of the trnsformtion opertor. On the other hnd, it should be pointed out tht since the eqution. hs rx piecewise continuous coefficient, the solution of this problem forms the integrl representtion not opertor trnsformtion nd we use this integrl representtion for the solution of inverse problem of considered problem.,.. First of ll, we derive the liner integrl eqution by the kernel of the integrl representtion nd then we show tht this eqution hs unique solution. Finlly, we prove the uniqueness theorem of inverse problem. Now, we will refer to the sequences {λ n } nd { }, n Z s the spectrl dt of the boundry vlue problem.,.. Consider the functions nd F x, y [ sin λn x cos λ n x ϕ T y, λ n µπ sin λ n x cos λ nx ϕ T y, λ n 6 F x, y F µx, y. 7 Then, it is obtined from 3. nd 3. tht [ F x, y ϕ x, λ n ϕ T y, λ n µπ ϕ x, λ nϕ T y, λ n. 8 Theorem 3.. The following liner integrl eqution nmed by Gelfnd-Levitn-Mrchenko type eqution is stisfied for ech fixed x, π by the kernel Gx, y of the integrl representtion.: Gx, µy F x, y µx Proof. It cn be written from. tht ϕ x, λ ϕx, λ It is obtined from. nd 3.5 tht Gx, sf s, yds, < y < x. 9 µx ϕx, λ n ϕ T y, λ n sin λy Gx, y cos λy ϕ x, λ n ϕ T y, λ n dy.

5 The Min Eqution of Inverse Problem for Dirc Opertors 63 µx Gx, s N ϕx, λ n ϕ T y, λ n ϕx, λ n µy sin λn s cos λ n s Then, ccording to these equlities, we cn write Ψ N x, y Φ N x, y Φ N x, y ϕ T y, λ n ds, ϕx, λ n ϕ T y, λ n sin λ n s, cos λ n s G T y, sds. Ψ N x, y Φ N x, y Φ Nx, y Φ Nx, y Φ N x, y, Φ Nx, y Φ N x, y µx [ ϕx, λ n ϕ T y, λ n µπ ϕx, λ nϕ T y, λ n, [ ϕ x, λ n ϕ T y, λ n µπ ϕ x, λ nϕ T y, λ n Gx, s µx µπ Gx, s [ sin λ n s µπ cos λ ns sin λ n s cos λ ns ϕx, λ n µy [ sin λn s cos λ n s ϕ T y, λ n ds,, ϕ T y, λ n ds, ϕ T y, λ n sin λ n s, cos λ n s G T y, sds. Now, let us exmine these expressions respectively. Assume tht gx, x [, π is bsolutely continuous function. Then, from Lemm.3, it is clculted uniformly with respect to x [, π tht Ψ N x, ygyrydy β n ϕx, λ n βnϕx, λ n. It follows from 3.3 tht Φ N x, ygyrydy [ ϕ x, λ n ϕ T y, λ n µπ ϕ x, λ nϕ T y, λ n gyrydy F x, ygyrydy. 3 It cn be written from. tht sin λs cos λs { ϕ s, λ, s <, ϕ s α α, λ, s >. 4

6 64 Ozge Akcy, Khnlr R. Mmedov Using.3 nd 3.9, we hve [ π Φ N x, ygyrydy [ µx Gx, s Gx, s [ αx α Gx, s Gx, sgsds sin λ n s µπ cos λ ns µπ ϕ s, λ nϕ T x, λ nds αx α ϕ T y, λ nds gyrydy gyrydy s µπ ϕ α α n, λ s Gx, sg α α ds. ϕ T x, λ nds gyrydy Substituting s α α η nd then chnging the denottion for integrtion vribles, we get It is found tht Φ N x, ygyrydy Gx, sgsds α Gx, ygydy α Gx, αη α gηdη Gx, αy α gydy Gx, µygyrydy. 5 Φ Nx, ygyrydy µx Gx, s [ sin λn s cos λ n s ϕ T y, λ n sin λ n s µπ cos λ ϕ T y, λ n gyrydsdy ns [ π µx Gx, sf s, yds gyrydy. 6 According to the expressions.5,.6 nd residue theorem, we hve Φ Nx, ygyrydy [ N [ N ϕx, λ n ϑx, λ n ωλ n µy µy sin λ n s, cos λ n s G T y, sds sin λ n s, cos λ n s G T y, sds gyrydy gyrydy

7 The Min Eqution of Inverse Problem for Dirc Opertors 65 [ N [ π ϑx, λ Res λλ n ωλ ϑx, λ πi I N ωλ [ πi µy µy I N ϑx, λ ωλ e Imλ µy sin λs, cos λs G T y, sds sin λs, cos λs G T y, sdsdλ gyrydy gyrydy µy e Imλ µy sin λs, cos λs G T y, sdsdλ gyrydy, 7 { λ : λ λ N π µπ }, N is sufficiently lrge number. Since s λ, the I N estimtes ϑ x, λ h cos λαπ x λ h sin λαπ x O e Imλ απ x, ϑ x, λ h sin λαπ x λ h cos λαπ x O e Imλ απ x, nd the following expressions [3, Lemm.3. µy mx λ y π e Imλ µy G i, y, s sin λsds, µy mx λ y π e Imλ µy G i, y, s cos λsds, i,, re vlid, it is obtined from.9 nd 3. tht Φ Nx, ygyrydy. 8 In this wy, it is clculted by using 3.6, 3.7, 3.8, nd 3.3 tht Gx, µygyrydy µx F x, ygyrydy Gx, sf s, ygyrydsdy. Then, in view of the rbitrriness of gx, this yields Gx, µy F x, y µx Gx, sf s, yds, < y < x. Theorem 3.. The eqution 3.4 hs unique solution Gx,. L, µx for ech fixed x, π. Proof. In the cse of x <, due to rx, this theorem is proved in [. Now, ssume tht < x. Then, the eqution 3.4 cn be rewritten s T x Gx,. K x Gx,. F x,., { gy, y < x, T x g y gαy α, < y x, K x g αx α gsf s, yds, < y < x. 9

8 66 Ozge Akcy, Khnlr R. Mmedov Let us obtin tht T x hs bounded inverse in L, π. Suppose tht T x g y φy, φy L, π nd φy for y > x. Using this nd 3.4, we cn write gy Tx φ { φy, y, y φ y α α, < y. Then, we clculte g L g y g y dy φ y φ y dy φ y α α φ y α dy α φ y φ y α α dy α φ y φ y dy c φ y φ y dy c φ L. Thus, we hve g L Tx φ L c φ L. Therefore, the opertor T x is invertible in L, π nd the min eqution 3.4 cn be expressed s follows Gx,. Tx K x Gx,. Tx F x,., Tx K x is completely continuous opertor in L, π. In tht cse, it suffices to show tht the homogeneous eqution tµy µx tsf s, yds 3 hs only trivil solution ty. Let ty be non-trivil solution of 3.5 nd ty for y x, π. Then, from 3. nd 3.5, we hve t µy t µy rydy µx ts µπ [ sin λn s ϕ T cos λ n s y, λ n ϕ T y, λ n t T µyrydsdy. sin λ n s cos λ ns In this equlity, using 3.9 we get t µy t µy rydy ts ts αx α αx α ϕ s, λ n ϕ T y, λ n t T µyrydsdy µπ ϕ s, λ nϕ T y, λ nt T µyrydsdy ts ts s ϕ α α, λ n ϕ T y, λ n t T µyrydsdy s µπ ϕ α α n, λ ϕ T y, λ nt T µyrydsdy.

9 Substituting s α α s, we find The Min Eqution of Inverse Problem for Dirc Opertors 67 t µy t µy rydy α α ts ts tαs α tαs α ϕ s, λ n ϕ T y, λ n t T µyrydsdy µπ ϕ s, λ nϕ T y, λ nt T µyrydsdy t µy t µy rydy tµs tµs ϕ s, λ n ϕ T y, λ n t T µyrydsdy µπ ϕ s, λ nϕ T y, λ nt T µyrydsdy ϕ s, λ n ϕ T y, λ n t T µyryrsdsdy µπ ϕ s, λ nϕ T y, λ nt T µyryrsdsdy. 3 In the expression 3.6, using Prsevl equlity tµy x tµyϕ y, λ µπ nrydy we obtin γ n tµyϕ y, λ n rydy. The system {φ y, λ n }, n Z is complete in L,r, π see [, therefore tµy, nmely T x t y. Since T x hs bounded inverse in L, π, we hve Gx,.. Consequently, the following theorem is proved by using Theorem 3. nd Theorem 3.: Assume tht LQx, h, h nd L Qx, h, h be two boundry vlue problems nd lso if certin symbol θ denotes n object relted to L, then the symbol θ denotes the corresponding object to relted L. Theorem 3.3. If λ n λ n,, n Z, then Qx Qx.e. on, π nd h h, h h. Nmely, the spectrl dt {λ n, }, n Z uniquely determines the boundry vlue problem.,.. Proof. Considering 3. nd 3., we hve F x, y F x, y nd F x, y F x, y. It is obtined from the min eqution 3.4 tht Gx, y Gx, y. The expression.3 implies tht Qx Qx.e. on, π. Now, ccording to order of precedence, respectively in the considertion of the equlities.,. nd.6, we clculte ϕx, λ n ϕx, λ n, ωλ n ωλ n nd κ n κ n. Finlly, h h nd h h re obtined by using. nd.5.

10 68 Ozge Akcy, Khnlr R. Mmedov Algorithm 3.. According to spectrl dt {λ n, }, n Z, the construction of the potentil function Qx is s follows: From the given numbers {λ n, }, n Z construct the functions F x, y nd F x, y respectively by the formuls 3. nd 3., Find the function Gx, y by solving the min eqution 3.4, Clculte Qx by the formul.3. R E F E R E N C E S [ M. J. Ablowitz, D. J. Kup, A. C. Newell nd H. Segur, Nonliner-evolution equtions of physicl significnce, Phys. Rev. Lett., 3973, 5-7. [ T. T. Dzbiev, The inverse problem for the Dirc eqution with singulrity, Dokl. Akd. Nuk Azerb. SSR, 966, No., 8-. [3 L. Fddeev nd L. Tkhtjn, Hmiltonin Methods in the Theory of Solitons, Springer, Berlin, 7. [4 G. Freiling nd V. Yurko, Inverse Sturm-Liouville Problems nd Their Applictions, Nov Science Publisher, Huntington, New York, 8. [5 M. G. Gsymov nd T. T. Dzbiev, Solution of the inverse problem by two spectr for the Dirc eqution on finite intervl, Dokl. Akd. Nuk Azerb. SSR, 966, No. 7, 3-6. [6 A. R. Ltifov, On the representtion of solution with initil conditions for Dirc equtions system with discontinuous coefficient, Proceeding of IMM of NAS of Azerbijn, 6, No. 4, [7 B. Y. Levin, Lectures on Entire Functions, Americn Mthemticl Society, Providence, 996. [8 B. M. Levitn, Inverse Sturm-Liouville Problems, VNU Science Press, Utrecht, The Netherlnds, 987. [9 S.G. Mmedov, The inverse boundry vlue problem on finite intervl for Dirc s system of equtions, Azerbidzn Gos. Univ. Ucen. Zp. Ser. Fiz-Mt. Nuk, 5975, [ Kh. R. Mmedov nd O. Akcy, Inverse eigenvlue problem for clss of Dirc opertors with discontinuous coefficient, Bound. Vlue Probl., 4:, 4, DOI:.86/ [ Kh. R. Mmedov nd Ö. Akçy, Inverse problem for clss of Dirc opertor, Tiwnese J. Mth., 84, No. 3, [ Kh. R. Mmedov nd O. Akcy, Necessry nd sufficient conditions for the solvbility of inverse problem for clss of Dirc opertors, Miskolc Mth. Notes, 65, No., [3 V. A. Mrchenko, Sturm-Liouville Opertors nd Applictions, AMS Chelse Publishing, Providence, Rhode Islnd,. [4 Y. V. Mykytyuk nd D. V. Puyd, Inverse spectrl problems for Dirc opertors on finite intervl, J. Mth. Anl. Appl. 386, No., [5 T. Sh. Abdullev nd I. M. Nbiev, An lgorithm for reconstructing the Dirc opertor with spectrl prmeter in the boundry condition, Comput. Mth. Mth. Phys., 566, No., [6 F. Prts nd J. S. Toll, Construction of the Dirc eqution centrl potentil from phse shifts nd bound sttes, Phys. Rev., 3 959, No., [7 D. V. Puyd, Inverse spectrl problems for Dirc opertors with summble mtrix-vlued potentils, Integrl Equtions Opertor Theory, 74, No. 3, [8 D. G. Shepelsky, An inverse spectrl problem for Dirc-type opertor with sewing, Dynmicl Systems nd Complex Anlysis, Nukov Dumk, Kiev, 99, pp. 4- Russin. [9 B. Thller, The Dirc Eqution, Springer, Berlin, 99. [ C.-F. Yng nd Z.-Y. Hung, Reconstruction of the Dirc opertor from nodl dt, Integrl Equtions Opertor Theory, 66, No. 4, [ C-F. Yng nd V. N. Pivovrchik, Inverse nodl problem for Dirc system with spectrl prmeter in boundry conditions, Complex Anl. Oper. Theory, 73, No. 4, -3.