ON TWO-SPECTRA INVERSE PROBLEMS

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1 ON TWO-SPECTRA INVERSE PROBLEMS NAMIG J GULIYEV Abstract We cosider a two-spectra iverse problem for the oe-dimesioal Schrödiger equatio with boudary coditios cotaiig ratioal Herglotz Nevalia fuctios of the eigevalue parameter ad provide a complete solutio of this problem Cotets 1 Itroductio ad mai result 1 Prelimiaries 3 3 Properties of two problems with a commo boudary coditio 6 4 Iverse problem 7 Appedix A Auxiliary results 9 Refereces 11 1 Itroductio ad mai result The study of two-spectra iverse problems was iitiated by Borg [4], who proved that the potetial q of the oe-dimesioal Schrödiger equatio y (x + q(xy(x = λy(x (11 is uiquely determied by the spectra of the boudary value problems geerated by this equatio ad the boudary coditios y (0 = h 1 y(0, y (π = Hy(π ad y (0 = h y(0, y (π = Hy(π respectively (with h 1 h Subsequet developmets by Marcheko [0], Krei [15], Levita ad Gasymov [17], [18] ad others showed that ot oly the potetial q but also the boudary coefficiets h 1, h ad H are uiquely determied by these spectra, ad that ay two iterlacig sequeces satisfyig certai asymptotic coditios are ideed the spectra of boudary value problems of the above form (see also [7], [1] These results were relatively recetly geeralized to problems with distributioal potetials [6], [13], [3] I this paper we are iterested i two-spectra iverse problems for boudary value problems with boudary coditios depedet o the eigevalue parameter Such problems have also bee cosidered i the literature Some uiqueess results were obtaied i [1], [], [3], [8] The papers [5], [19] cotai some existece results for problems with oe eigevalue-parameter-depedet boudary coditio I the case whe oly oe of the boudary coditios depeds liearly o the eigevalue parameter, ecessary ad sufficiet coditios for solvability of the two-spectra 010 Mathematics Subject Classificatio 34A55, 34B07, 34B4, 34L40, 47A75, 47E05 Key words ad phrases two-spectra iverse problem, oe-dimesioal Schrödiger equatio, boudary coditios depedet o the eigevalue parameter 1

2 NAMIG J GULIYEV iverse problem were foud i [16], [9] For problems with coupled boudary coditios depedet o the eigevalue parameter, see [14] ad the refereces therei We cosider two-spectra iverse problems for boudary value problems geerated by the equatio (11 together with boudary coditios of the form y (0 y(0 = f(λ, where q L (0, π is real-valued ad f(λ = h 0 λ + h + d k=1 y (π y(π δ k h k λ, F (λ = H 0λ + H + = F (λ, (1 D k=1 k H k λ (13 are ratioal Herglotz Nevalia fuctios with real coefficiets, ie, h 0, H 0 0, δ k, k > 0, h 1 < < h d, H 1 < < H D Usig Darboux-type trasformatios betwee such boudary value problems, we recetly obtaied i [11] various direct ad iverse spectral results for boudary value problems of the form (11, (1 But these trasformatios are ot applicable to two-spectra iverse problems because a pair of boudary value problems with a commo boudary coditio is trasformed to a pair of boudary value problems with o commo boudary coditios Therefore we first reduce our two-spectra problem to a iverse problem solved i [11] ad the completely solve the two-spectra problem We deote the boudary value problem (11-(1 by P(q, f, F, ad assume that α 0 ad id f 0 so that the problems P(q, f, F ad P(q, f + α, F are differet We use the otatio ( 1 x = y + l to mea =0 (x y < We also assume that o eigevalue of P(q, f, F is a pole of f or, which is the same, the spectra of the problems P(q, f, F ad P(q, f + α, F do ot itersect It turs out that the problems P(q, f, F ad P(q, f + α, F are ot, i geeral, uiquely determied by their spectra However, we are able to describe all pairs of problems with the give two spectra Our mai result reads as follows Theorem 11 Two sequeces {λ } 0 ad {µ } 0 are the eigevalues of a pair of problems of the form P(q, f, F ad P(q, f + α, F if ad oly if they iterlace ad satisfy asymptotics of the form λ = L + σ π + l ( 1, λ ( ( 1 µ = ( L r 1 ν + l for some iteger or half-iteger L 1/, σ R, ν R \ {0} ad r {0, 1}, with the exceptio of the case whe L = 1/ ad r = 1 Moreover, there is a oe-to-oe correspodece betwee such problems ad sets of oegative itegers of cardiality ot exceedig L + (1 r/ I particular, the proof of this theorem (Sectio 4 also yields the followig uiqueess result Corollary 1 The problems P(q, f, F ad P(q, f +α, F are uiquely determied by their spectra ad the poles of f

3 ON TWO-SPECTRA INVERSE PROBLEMS 3 So i a sese, the amout of spectral data required for the uique determiatio i our case (two spectra ad a fiite umber of idices is betwee that of the classical case (two spectra oly ad of problems with coupled boudary coditios (two spectra ad a ifiite sequece of sigs; for the latter case see, eg, [1] ad the refereces therei The paper is orgaized as follows I Sectio we itroduce the ecessary otatio ad obtai some useful idetities I Sectio 3 we fid some coditios for the eigevalues of the problems P(q, f, F ad P(q, f + α, F Sectio 4 is devoted to the proof of our mai result, Theorem 11 Fially, i the appedix we prove two auxiliary lemmas used i the mai text Prelimiaries First we itroduce some further otatio We assig to each fuctio f of the form (13 two polyomials f ad f by writig this fuctio as where f (λ := h 0 We defie the idex of f as If f = the we just set f(λ = f (λ f (λ, d (h k λ, h 0 := k=1 id f := deg f + deg f { 1/h 0, h 0 > 0, 1, h 0 = 0 f (λ := 1, f (λ := 0, id f := 1 It ca easily be verified that each ocostat fuctio f of the form (13 is strictly icreasig o ay iterval ot cotaiig ay of its poles, ad f(λ ± (respectively, f(λ h as λ ± if its idex is odd (respectively, eve Let ϕ(x, λ, ψ(x, λ ad χ(x, λ be the solutios of (11 satisfyig the iitial coditios ϕ(0, λ = f (λ, ψ(0, λ = f (λ, χ(π, λ = F (λ, ϕ (0, λ = f (λ, ψ (0, λ = f (λ αf (λ, χ (1 (π, λ = F (λ The the eigevalues of the boudary value problems P(q, f, F ad P(q, f + α, F coicide with the zeros of (their characteristic fuctios ad Φ(λ := F (λϕ(π, λ F (λϕ (π, λ = f (λχ (0, λ + f (λχ(0, λ Ψ(λ := F (λψ(π, λ F (λψ (π, λ = f (λχ (0, λ + (f (λ + αf (λ χ(0, λ respectively details ad These eigevalues have the asymptotics (see [11, Theorem 41] for λ = µ = id f + id F id f + id F + σ π + l + σ π + l ( 1 ( 1 (

4 4 NAMIG J GULIYEV with σ σ = { α, id f is eve 0, id f is odd I the ext sectio we will obtai more refied asymptotics for the differece of the square roots of the eigevalues λ ad µ Sice for each eigevalue λ of P(q, f, F the solutios ϕ(x, λ ad χ(x, λ are liearly depedet, there exists a uique umber β 0 such that χ(x, λ = β ϕ(x, λ (3 The ormig costats of the problem P(q, f, F are defied as γ := π 0 ϕ (x, λ dx + f (λ f (λ + F (λ F (λ F (λ F (λ β They have the asymptotics ([11, Theorem 41] γ = π ( id f + id F id f ( 1 + l ( 1 (4 The sequeces {λ } 0, {β } 0 ad {γ } 0 are related by the idetity ([11, Lemma 1] Φ (λ = β γ (5 I the remaiig part of this sectio, we are goig to show that the coefficiets of the polyomial f (λ satisfy a osigular system of liear equatios whose coefficiets are expressed i terms of the sequeces {λ } 0 ad {γ } 0 Thus ay polyomial whose coefficiets satisfy this system must ecessarily coicide with f (λ We will eed this result i Sectio 4 We start with some idetities for the eigevalues ad the ormig costats of the problem P(q, f, F Such idetities are characteristic to problems with boudary coditios depedet o the eigevalue parameter; they were used i [10] to obtai explicit expressios for all the coefficiets of the boudary coditios i the case of liear depedece o the eigevalue parameter (ie, id f = id F = i our otatio Lemma 1 The followig idetities hold: λ k f (λ = 0, k = 0,, d 1 γ =0 Proof From (1 ad (3 we have =0 f (λ = ϕ(0, λ = χ(0, λ β Together with (5 this implies (for sufficietly large N N λ k f (λ N λ k χ(0, λ = Res λ=λ = 1 λ k χ(0, λ γ Φ(λ πi C N Φ(λ =0 where C N is the circle of radius ( N id f + id F 1 dλ,

5 ON TWO-SPECTRA INVERSE PROBLEMS 5 cetered at the origi Expressig χ(x, λ as( a liear combiatio of the cosiead sie-type solutios we obtai χ(x, λ = O λ e id F Im λπ O the other had, from (A1 we get (see, eg, the proof of [7, Theorem 113] for details ( 1 Φ(λ = O (id f+id F +1 λ e Im λπ C N,, λ N ad thus λ k χ(0, λ Φ(λ with id f k Hece which proves the lemma ( = O lim N 1 N id f k+1 λ k χ(0, λ C N Φ(λ, λ C N N dλ = 0, Deote by p d 1,, p 0 the o-leadig coefficiets of the polyomial f (λ after dividig it by its leadig coefficiet: ( 1 d f (λ = h 0 d (λ h k = λ d + p d 1 λ d p 1 λ + p 0 k=1 It is easy to see from the asymptotics of the eigevalues ad the ormig costats that for each k = 0,, d 1 the series λ k s k := γ =0 coverges absolutely Lemma 1 implies the followig idetities betwee the umbers p i ad s j : d 1 p i s i+k = s d+k, k = 0, 1,, d 1 (6 i=0 We cosider them as a system of liear equatios (with respect to the umbers p i, the matrix of which is the followig Hakel matrix: s 0 s 1 s d 1 s 1 s s d s d 1 s d s d The quadratic form correspodig to this matrix is positive defiite: d 1 i,j=0 s i+j ξ i ξ j = d 1 i,j=0 =0 λ i+j ξ i ξ j = γ d 1 =0 i,j=0 λ i+j ξ i ξ j = γ 1 γ =0 ( d 1 λ i ξ i 0 with equality if ad oly if d 1 i=0 λi ξ i = 0 for all, ie ξ 0 = = ξ d 1 = 0 Thus the determiat of the above matrix is strictly positive ad hece the system (6 has a uique solutio i=0

6 6 NAMIG J GULIYEV 3 Properties of two problems with a commo boudary coditio We are ow goig to study further properties of the sequeces {λ } 0 ad {µ } 0 We will first show that these two sequeces iterlace ad the fid more refied asymptotics for the differece of their square roots As we will see i the ext sectio, ay two sequeces with these two properties are ideed the eigevalues of a pair of boudary value problems with a commo boudary coditio The fuctio m(λ := Ψ(λ Φ(λ satisfies the idetity m(λ = m(λ ad is a meromorphic fuctio with poles at λ ad zeros at µ For oreal values of λ the solutio satisfies the boudary coditio 0 y(x, λ := ψ(x, λ + m(λϕ(x, λ F (λy(π, λ F (λy (π, λ = 0 Usig (1 we calculate π (λ µ y(x, λy(x, µ dx = (y(x, λy (x, µ y (x, λy(x, µ For µ = λ this yields = (F (µ F (λ y(π, λy(π, µ + αf (λf (µ (m(λ m(µ + (f (λf (µ f (µf (λ (1 + m(λ (1 + m(µ π Im m(λ 1 α = Im λ f (λ y(x, λ dx 0 + y(π, λ Im F (λ f (λ Im λ Im f(λ m(λ Im λ > 0 Thus αm(λ is a Herglotz Nevalia fuctio, ad hece its zeros µ ad poles λ iterlace Usig (1, (3 ad the costacy of the Wroskia we obtai Ψ(λ = F (λ ψ(π, λ F (λ ψ (π, λ = β (ϕ (π, λ ψ(π, λ ϕ(π, λ ψ (π, λ = β (ϕ (0, λ ψ(0, λ ϕ(0, λ ψ (0, λ = αβ f (λ Together with (5 this implies γ = αf (λ Φ (λ Ψ(λ We will eed this formula i the ext sectio i order to trasform our two-spectra iverse problem to a iverse problem solved i [11], but for ow we will use it to obtai more refied asymptotics for the differece λ µ The mea value theorem yields ( Ψ(λ = Ψ(λ Ψ(µ = λ ( µ λ + µ Ψ (ζ (31 π 0

7 ON TWO-SPECTRA INVERSE PROBLEMS 7 for ζ [λ, µ ] with ζ = (id f + id F / + O ( 1 Thus λ αf µ = (λ Φ (λ ( λ + µ γ Ψ (ζ Applyig Lemma A1 to the problems P(q, f, F ad P(q, f +α, F, ad the applyig Lemma A3 to the fuctios Φ ad Ψ ad usig (4, we obtai the asymptotics λ ( r 1 id f + id F (α (h µ = 0 ( 1 + l π, where r := id f d = { 1, id f is odd, 0, id f is eve 4 Iverse problem I this sectio, we will prove Theorem 11 The results of the previous sectio shows that if two sequeces {λ } 0 ad {µ } 0 are the eigevalues of the problems P(q, f, F ad P(q, f + α, F, the they iterlace ad satisfy asymptotics of the form λ = L + σ π + l ( 1, λ ( ( 1 µ = ( L r 1 ν + l (41 where id f + id F L := d 1 r 1, ν := α (h 0 0 π Note also that if L = 1/ the id f = 0, ad cosequetly r = 0 We are ow goig to prove that these coditios are also sufficiet for two sequeces to be the eigevalues of two such problems However, ulike the case of costat boudary coditios, i order to determie these problems uiquely, we eed some additioal data The idetity Ψ(λ Φ(λ = αf (λχ(0, λ (see Sectio shows that the zeros of f are also zeros of Ψ Φ As we will see shortly, they ca be chose arbitrarily amog the zeros of Ψ Φ Let ow {λ } 0 ad {µ } 0 be ay two sequeces such that they iterlace ad satisfy asymptotics of the form (41 for some iteger or half-iteger L 1/, real ν 0 ad σ, ad r {0, 1} Defie the fuctios ad Φ(λ := <L Ψ(λ := <L Let d be a iteger with (λ λ =L (µ λ =L π(λ λ >L π(µ λ >L λ λ ( L µ λ ( L 0 d L + 1 r, ad let i 1, i,, i d be itegers (idices with 0 i 1 < i < < i d Defie the polyomial d p(λ := (τ ik λ, k=1

8 8 NAMIG J GULIYEV where τ 0 < τ 1 < are the zeros of the fuctio Φ(λ Ψ(λ Lemma A3 implies ( ( Φ (λ = ( 1 ( L L π 1 + l Usig (31 ad Lemma A3 agai we also have ( ( 1 Ψ(λ = ( 1 ( L L r πν + l From the asymptotics of λ µ ad the fact that λ ad µ iterlace it follows that if ν > 0 (respectively, ν < 0 the µ < λ < µ +1 (respectively, λ < µ < λ +1 for each 0 Thus the umbers γ defied by are all positive ad have the asymptotics γ := πνp (λ Φ (λ Ψ(λ γ = ( L 4d+r ( π + l ( 1 By [11, Theorem 44], there exists a boudary value problem P(q, f, F havig the eigevalues {λ } 0 ad the ormig costats {γ } 0 Moreover, id f = d + r 0 ad id F = L d r 1 Deote α := πν/ (h 0 with h 0 defied as at the begiig of Sectio It oly remais to show that the problem P(q, f + α, F has the eigevalues µ But first we show that the polyomials f (λ ad p(λ coicide up to a costat factor Arguig as i the proof of Lemma 1 we have =0 λ k p(λ γ = =0 λ k Ψ(λ πνp(λ Φ (λ = 1 π νi λ k (Ψ(λ Φ(λ lim dλ = 0, N C N p(λφ(λ where C N is the same as i that proof Now arguig as after Lemma 1 we obtai that the o-leadig coefficiets of the polyomial ( 1 d p(λ satisfy the system (6 Therefore f (λ = h 0p(λ Deote the eigevalues of the boudary value problem P(q, f + α, F by µ They coicide with the zeros of the fuctio Ψ(λ := F (λψ(π, λ F (λψ (π, λ, where ψ(x, λ is defied as i (1 Usig the results of Sectio 3, we obtai Ψ(λ = αf (λ Φ (λ γ = πνp (λ Φ (λ γ = Ψ(λ, 0 This ad the proofs of Lemmas 1, A1 ad A3 show that is a etire fuctio satisfyig the estimate ( 1 λ Ψ(λ Ψ(λ Φ(λ = O ( Ψ(λ Ψ(λ /Φ(λ o N C N ad hece by the maximum priciple o the whole plae The the Liouville theorem implies that this fuctio is idetically zero Thus Ψ(λ Ψ(λ ad hece µ = µ, 0

9 ON TWO-SPECTRA INVERSE PROBLEMS 9 Appedix A Auxiliary results I this appedix we prove two auxiliary lemmas used i the mai body of the paper Lemma A1 The characteristic fuctio Φ(λ of P(q, f, F with id f 0 has the ifiite product represetatio Φ(λ = <L (λ λ =L π(λ λ >L λ λ ( L with L := Proof The first-order asymptotics Φ(λ = λ L+1/ si id f + id F ( ( λ + L π + O λ L e Im λπ (A1 was obtaied i [11] (see the proof of Lemma therei From Hadamard s theorem we obtai Φ(λ = C (1 λλ = C ( 1 λ ( 1 λ (1 λλ λ =0 λ <L =L >L Accordig to our assumptio, L is a iteger or half-iteger with L 1/ The we ca combie ifiite product represetatios for the sie ad cosie fuctios ito The use of the idetities ( si λ + L π = ( 1 L π λ ( 1 =L >L ( 1 L = ( 1 <L =L ( 1, λ L+1/ = <L λ ( L λ =L λ yields ( λ L+1/ si λ + L π = ( λ ( πλ ( λ 1 ( L <L =L >L Thus Φ(λ ( λ = C ( 1 1 ( 1 1 λ L+1/ si + L π λ λ πλ πλ <L =L ( ( L 1 + λ ( L λ ( L λ >L >L Takig the limit as λ ad usig (A1 ad ( we obtai C = λ λ πλ ( L, <L =L >L which proves the lemma To prove our ext result we eed a lemma of Marcheko ad Ostrovskii

10 10 NAMIG J GULIYEV Lemma A ([, Lemma 33], [1, Lemma 34] For fuctios u(z ad v(z to admit represetatios of the form 4z u(z = si πz + Aπ 4z 1 cos πz + g 1(z z si πz, v(z = cos πz Bπ z + g (z, z where g 1 (z = π 0 g 1(t cos zt dt ad g (z = π 0 g (t si zt dt with g 1, g L [0, π] ad π 0 g 1(t dt = 0, it is ecessary ad sufficiet to have the form v(z = u(z = πz =1 Now we ca prove (u z, u = A + l =1 ( 1, ( 1 (v z, v = 1 B + l ( 1 Lemma A3 Let {η } 0 ad {ζ } 0 be sequeces of real umbers havig the asymptotics η = L + σ ( ( 1 π + l 1, ζ = L + O, with a iteger or half-iteger L 1/ ad a real σ, ad let The G(λ := <L (η λ =L π(η λ >L G (ζ = ( 1 ( L L ( π + l η λ ( L ( 1 Proof Usig the asymptotics of η ad Lemma A we obtai the represetatios ( ( G(λ = λ L+1/ si λ + L π σλ L cos λ + L π + G 1 (λ ad G (λ = π λl cos ( ( λ + L π + L + σπ + 1 ( λ L 1/ si λ + L π + G (λ, where G 1 (λ ad G (λ are of the form ad G 1 (λ = λ L π G (λ = λ L 1/ π 0 ( G 1 (t cos λt + Lπ dt + O (λ L 1/ e Im λπ 0 ( G 1 (t si λt + Lπ dt + O (λ L 1 e Im λπ with G 1 L [0, π] The statemet of the theorem ow follows from the asymptotics of ζ

11 ON TWO-SPECTRA INVERSE PROBLEMS 11 Refereces [1] R Kh Amirov, A S Ozka ad B Keski, Iverse problems for impulsive Sturm Liouville operator with spectral parameter liearly cotaied i boudary coditios, Itegral Trasforms Spec Fuct 0 (009, o 7-8, [] A I Beedek ad R Pazoe, O iverse eigevalue problems for a secod-order differetial equatio with parameter cotaied i the boudary coditios, Uiversidad Nacioal del Sur, Istituto de Matemática, Bahía Blaca, 1980 [3] P A Bidig, P J Browe ad B A Watso, Equivalece of iverse Sturm Liouville problems with boudary coditios ratioally depedet o the eigeparameter, J Math Aal Appl 91 (004, o 1, [4] G Borg, Eie Umkehrug der Sturm Liouvillesche Eigewertaufgabe, Acta Math 78 (1946, 1 96 [5] M V Chuguova, Iverse spectral problem for the Sturm Liouville operator with eigevalue parameter depedet boudary coditios, Operator theory, system theory ad related topics, Birkhäuser, Basel, 001, pp [6] J Eckhardt, F Gesztesy, R Nichols ad G Teschl, Iverse spectral theory for Sturm Liouville operators with distributioal potetials, J Lod Math Soc ( 88 (013, o 3, arxiv: [7] G Freilig ad V Yurko, Iverse Sturm Liouville problems ad their applicatios, Nova Sciece Publishers, Ic, Hutigto, NY, 001 [8] G Freilig ad V Yurko, Iverse problems for Sturm Liouville equatios with boudary coditios polyomially depedet o the spectral parameter, Iverse Problems 6 (010, o 5, , 17 pp [9] N J Guliyev, Iverse eigevalue problems for Sturm Liouville equatios with spectral parameter liearly cotaied i oe of the boudary coditios, Iverse Problems 1 (005, o 4, arxiv: [10] N J Guliyev, A uiqueess theorem for Sturm Liouville equatios with a spectral parameter liearly cotaied i the boudary coditios, Proc Ist Math Mech Natl Acad Sci Azerb 5 (006, [11] N J Guliyev, Essetially isospectral trasformatios ad their applicatios, preprit arxiv: [1] I M Guseiov ad I M Nabiev, Solutio of a class of iverse Sturm Liouville boudary value problems (Russia, Mat Sb 186 (1995, o 5, 35 48; Eglish trasl i Sb Math 186 (1995, o 5, [13] R O Hryiv ad Ya V Mykytyuk, Iverse spectral problems for Sturm Liouville operators with sigular potetials II Recostructio by two spectra, Fuctioal aalysis ad its applicatios, Elsevier, Amsterdam, 004, pp arxiv:math/ [14] Ch G Ibadzadeh ad I M Nabiev, Recostructio of the Sturm Liouville operator with oseparated boudary coditios ad a spectral parameter i the boudary coditio (Russia, Ukraï Mat Zh 69 (017, o 9, ; Eglish trasl i Ukraiia Math J 69 (018, o 9, [15] M G Krei, Solutio of the iverse Sturm Liouville problem (Russia, Doklady Akad Nauk SSSR (NS 76 (1951, 1 4 [16] N Dzh Kuliev, Iverse problems for the Sturm Liouville equatio with a spectral parameter i the boudary coditio (Russia, Dokl Nats Akad Nauk Azerb 60 (004, o 3-4, [17] B M Levita, O the determiatio of a Sturm Liouville equatio by two spectra (Russia, Izv Akad Nauk SSSR Ser Mat 8 (1964, o 1, 63 78; Eglish trasl i Amer Math Soc Trasl ( 68 (1968, 1 0 [18] B M Levita ad M G Gasymov, Determiatio of a differetial equatio by two of its spectra (Russia, Uspehi Mat Nauk 19 (1964, o (116, 3 63; Eglish trasl i Russia Math Surveys 19 (1964, o, 1 63 [19] S G Mamedov, Determiatio of a secod-order differetial equatio from two spectra with a spectral parameter eterig ito the boudary coditios (Russia, Izv Akad Nauk Azerbaĭdzha SSR Ser Fiz-Tekh Mat Nauk 3 (198, o, 15

12 1 NAMIG J GULIYEV [0] V A Marcheko, Some questios of the theory of oe-dimesioal liear differetial operators of the secod order I (Russia, Trudy Moskov Mat Obšč 1 (195, 37 40; Eglish trasl i Amer Math Soc Trasl ( 101 (1973, [1] V A Marcheko, Sturm Liouville operators ad applicatios (Russia, Naukova Dumka, Kiev, 1977; Eglish trasl: AMS Chelsea Publishig, Providece, RI, 011 [] V A Marcheko ad I V Ostrovskii, A characterizatio of the spectrum of the Hill operator (Russia, Mat Sb (NS 97(139 (1975, o 4(8, ; Eglish trasl i Math USSR- Sb 6 (1975, o 4, [3] A M Savchuk ad A A Shkalikov, Iverse problem for Sturm Liouville operators with distributio potetials: recostructio from two spectra, Russ J Math Phys 1 (005, o 4, Istitute of Mathematics ad Mechaics, Azerbaija Natioal Academy of Scieces, 9 B Vahabzadeh str, AZ1141, Baku, Azerbaija address: jguliyev@gmailcom