THE CHARACTERIZATION PROBLEM FOR ONE CLASS OF SECOND ORDER OPERATOR PENCIL WITH COMPLEX PERIODIC COEFFICIENTS

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1 MOSCOW MATHEMATICAL JOURNAL Volume 7, Number 1, January March 2007, Pages THE CHARACTERIZATION PROBLEM FOR ONE CLASS OF SECOND ORDER OPERATOR PENCIL WITH COMPLEX PERIODIC COEFFICIENTS R. F. EFENDIEV Absrac. The purpose of he presen work is solving he characerizaion problem, which consiss of idenificaion of necessary and sufficien condiions on he scaering daa ensuring ha he reconsruced poenial belongs o a paricular class Mah. Subj. Class. 34B25, 34L05, 34L25, 47A40, 81U40. Key words and phrases. Inverse problem, characerizaion problem, scaering daa, ransformaion operaor. 1. Inroducion The purpose of he presen work is solving he characerizaion problem, which consiss of idenificaion of necessary and sufficien condiions on he scaering daa ensuring ha he reconsruced poenial belongs o a paricular class. In our case Q 2 is he class of all 2π periodic complex valued funcions on he real axis R, belonging o L 2 [0, 2π] and Q 2 + is is subclass consising of funcions qx = q n expinx. 1.1 The objec under consideraion is he operaor L generaed by he differenial expression d l dx, λ d2 dx 2 + 2λpx + qx λ2 1.2 in he space L 2, wih poenials px = p ne inx, qx = q ne inx, for which n p n <, q n < are fulfilled, λ is a specral parameer. The inverse problem for he poenials 1.1 was formulaed and solved in he papers [3], [4], where i was shown, ha he equaion Ly = 0, has he soluion e ± x, λ = e 1 ±iλx + V n ± e inx V nα ± + n ± 2λ eiαx, 1.3 α=n and Wronskian of he sysem of soluions e ± x, λ being equal o 2iλ. Received November 3, c 2007 Independen Universiy of Moscow

2 56 R. EFENDIEV The limi e ± n x = lim n ± 2λe ±x, λ = λ n/2 V nαe ± iαx e i n 2 x, n N, is also he soluion of he equaion ly = 0, bu already linearly dependen on e ± x, ± n 2. Therefore, here exis he numbers Ŝn, n N, for which he condiions e ± n x = Ŝ± n e x, n, n N, are fulfilled. From he las relaion one may obain ha Ŝ± n = V nn. ± In [3] he specral analysis of he operaor pencil L was carried ou and sufficien condiion for reconsrucion of px, qx Q 2 + using he values Ŝn, n N, was found. Noe ha some of he characerizaions for he Surm Liouville operaor wih he real-valued poenials belonging o L 1 1R L 1 αr is he class of measurable poenials saisfying he condiion R dx1 + x α p γ x <, have been given by Melin [9] and Marchenko [8]. More deails review can be found in he papers [1], [2], [7]. For he poenials px = 0, qx Q 2 +, which in he nonrivial cases are complex valued, he inverse problem was firs formulaed and solved by Gasymov [5]. Laer he complee soluion of he inverse problem for he cases px = 0, qx Q 2 + was found by Pasur and Tkachenko [10]. Now le us formulae he basic resul of he presen work. Definiion. The sequence {Ŝ± n } consruced by means of he formulae 1.4, is called a se of specral daa of he operaor 1.2 wih poenials px, qx Q 2 +. Theorem 1. For a given sequence of complex numbers {Ŝ± n } o be a se of specral daa of he operaor L generaed by he differenial expression 1.2 and poenials px, qx Q 2 + i is necessary and sufficien ha he following condiions are fulfilled: 1 α=n {n 2 Ŝ ± n } l 1 ; Infinie deerminan Dz de δ nm 4Ŝ mŝ+ m+k k m + kn + k ei 2 z e i n+k 2 z n,m=1 1.6 exiss, is coninuous, no equal o zero in he closed half-plane C + = {z: Im z 0} and analyical inside of he open half-plane C + = {z: Im z > 0}. 2. On an Inverse Problem of he Scaering Theory on he Semiaxis On he base of he proof of Theorem 1, we will sudy he equaion Ly = 0. Denoing x = i, λ = iµ, yx = Y 2.1

3 OPERATOR PENCILS WITH COMPLEX PERIODIC COEFFICIENTS 57 we obain he equaion in which p = ipi = i Y + 2µ piy + qiy = µ 2 Y 2.2 p n e n, q = qi = q n e n. 2.3 As a resul we obain he equaion 2.2, whose poenials exponenially decrease as. The specificaion of he considered inverse problem is defined by he fac ha he poenials belong o he class Q 2 +. In his secion we suppose R +. The procedure of analyic coninuaion ha allows o ge corresponding resuls for he equaion 1.2 from he resul for he 2.2 will be invesigaed in he nex secion. The equaion 2.2 wih poenials 2.3 has he soluion α=n f ±, µ = e 1 ±iµ + V n ± e n V nα ± + in ± 2µ e α 2.4 and he numbers V n ±, V nα ± are defined by he following recurren formulae: α α 1 s α 2 V α ± + α V nα ± + q α s V s ± ± p α s + q α = 0, 2.5 s=1 V ± ns α 1 αα nv nα ± + q α s n p α s V ns ± = 0, 2.6 s=n α 1 αv α ± ± V s ± p α s ± p α = s=1 and he sequence 2.4 admis double ermwise differeniaion. Then wih he help of he condiion 2.3 we obain f ±, µ = Ψ ± e ±iµ + α=n K ±, ue ±iµu du, 2.8 where K ±, u, Ψ ± have he form K ±, u = 1 V ± 2i nαe α e u n/2, Ψ ± = 1 + V n ± e n, 2.9 So, i is proved he following Lemma 1. The funcion Ψ ± and he kernel of he ransformaion operaor of he equaion 2.2 K, u, u, aached o +, wih he poenials 2.3 permis he represenaion 2.9, in which he series n 2 V n ± 1 ; αα n V ± n nα ; n V nn ± are convergen. α=n+1

4 58 R. EFENDIEV Remark. In our case he kernel of he operaor of ransformaion K ±, u, u, a +, and he funcion Ψ ± are consruced effecively. Then i is possible [3] o ge he equaliy f n ± = S n ± f, i n, where f ± n = lim in ± 2µf ±, µ. µ in/2 Rewriing he equaliy 2.10 in he form V nαe ± α e n/2 = S n ± e 1 n/2 + Vm e m + α=n and denoing by z ± + s = we obain he Marchenko ype equaion m=1 m=1 α=m V mα im + n e α 2.11 S me ± +sm/ m=1 K ±, s = Ψ ± z ± + s + So, i is proved he following K, uz ± u + s du Lemma 2. If he coefficiens p and q of he equaion 2.2 have he form 2.3, hen a every 0, he kernel of he ransformaion operaor 2.9 saisfies o he equaion of he Marchenko ype 2.13 in which he ransiion funcion z ± has he form 2.12 and he numbers S ± m are defined by he equaliy 2.10, from which i is obained, ha S ± m = V ± mm. Noe ha from relaion 2.7 one may easily obain he formulae Ψ + Ψ = 1 and lim x Ψ ± x = 1 useful laer on. The poenials are reconsruced by he kernel of he ransformaion operaor and he funcion Ψ ± wih he help of he formulae Ψ ± = J ± i puψ ± u du, 2.14 K ±, = ± 1 2 quψ ± u du i pψ ± ± i + puk ± u, u du 2.15 Hence he basic equaion 2.13 and he form of he ransiion funcion 2.12 make naural he formulaion of he inverse problem abou reconsrucion of he poenials of he equaion 2.2 by numbers S ± n. In his formulaion, which employs he ransformaion operaor, an imporan momen is a proof of unique solvabiliy of he basic equaion Lemma 3. The homogenous equaion g ± s 0 z ± u + sg u du = 0, 2.16

5 OPERATOR PENCILS WITH COMPLEX PERIODIC COEFFICIENTS 59 corresponding o he poenial p, q Q 2 + has only rivial soluion in he space L 2 R +. The proof of Lemma 3 is similar o [7, p. 198]. Lemma 4. A every fixed value a, Im a 0, he homogenous equaion g ± s has only rivial soluion in he space L 2 R +. z ± u + s 2aig u du = 0, 2.17 Proof. We subsiue x + a for x, where Im a 0 in equaion 1.2, and we obain he same equaion wih he coefficiens p α x = p α x + a, q α x = q α x + a belonging o Q 2 +. Le us remark, ha he funcions e ±x + a, λ are soluions of he equaion ha as x have he form y + 2λp a xy + q a xy = λ 2 y e ± x + a, λ = e ±iaλ e ±iλx + o1. Therefore, he funcions e a ±x, λ = e iaλ e ± x + a, λ are also soluions of ype 1.3. Then le us denoe by {Ŝ± n a} he specral daa of he operaor L wih he poenials p α x, q α x According o 1.4, we have L d2 dx 2 + 2λp ax + q a x λ 2. Ŝn aea ± x, ±n/2 = lim n λ ±n/2 2λea x, λ = Hence = lim λ ±n/2 n 2λe±iaλ e x + a, λ = e ian/2 Ŝ n e ± x + a, ±n/2 = = e ian/2 Ŝ n e ian/2 e a ±x, ±n/2 = Ŝ n e ian e a ±x, ±n/2 Ŝ n a = Ŝ n e ian. Now arguing as above, we obain he basic equaion of he form 2.13 wih he ransiion funcion Z a ± = S n ± ae n/2 = S n ± e ian e n/2 = Z ± 2ia. The nex heorem follows from Lemmas 3 and 4. Theorem 2. The poenials p and q of he equaion 2.2, saisfying he condiion 2.3 are uniquely defined by he numbers S ± n.

6 60 R. EFENDIEV 3. Proof of Theorem 1 Necessiy: The necessiy of he condiion 1 is proved in [3]. To prove he necessiy of he condiion 2 of he Theorem 1 le us demonsrae firs of all, ha from he rivial solvabiliy of he basic equaion 2.13 a = 0 in he class of funcions saisfying o he inequaliy gu Ce u 2, u 0, i follows rivial solvabiliy in l 2 R + of he infinie sysem of equaions or g ± n g ± n m=1 m=1 2S ± m m + n g m = S ms ± k m + kn + k g± m = 0, 3.2 where g n ± l 2 R; S n ± l 1. Really, if {g n } l 2 is a soluion of his sysem, hen he funcion g ± 2S u = ms ± k m + k e ku/2 g m ± 3.3 m=1 is defined for all u 0, saisfies he inequaliy and i is a soluion of equaion 2.16 g ± s m=1 0 0 g ± u c e u/2, u 0, z ± u + sz u + s 1 g ± s 1 ds 1 du = 2S ms ± k m + k e ks/2 g ± m + r= S n S ± r n + r e rs 1/2 g ± n m=1 = 2S ms ± k m + k e ks/2 m=1 0 m=1 ds 1 du = 2S ms ± k m + k e ks/2 r=1 [ SmS ± k e u+sk/2 e u+s1m/2 g ± m m=1 2SmS ± k m + k e ks/2 g m ± + 4S n S ± r n + rm + r g± n r=1 = 2S ± r S n m + rn + r g± n Since g ± u = 0 hen, SmS ± k g± m = 0 for all m 1, k 1, and g m ± = 0, m 1 according o 3.2. Le us inroduce in he space l 2 operaor F 2 ±, given by he marix F mn ±2 4Sn S ± k = n + km + k e m+k/2 e n+k/2, n, m N. 3.4 Then, we obain from n 2 S n ± l 1, ha j, F 2 ± ϕ j, ϕ k l2 <, i. e. F is a kernel operaor [11]. Therefore, here exiss he deerminan ± = ] = 0.

7 OPERATOR PENCILS WITH COMPLEX PERIODIC COEFFICIENTS 61 dee F 2 ± of he operaor E F 2 ± conneced, as easy o see, wih he deerminan D ± z from he condiion 2 of Theorem 1, wih relaion ± iz = dee F 2 ± iz D± z. The deerminan of sysem 3.1 is D ± 0, and he deerminan of he similar sysem corresponding o he poenials p z x = px+z, q z x = qx+z, Im z > 0 is D ± z = de δ 4Sn zs ± k mn z m + kn + k m, = δ 4Sn S ± m+k k mn m + kn + k ei 2 z e i n+k 2 z. m, Therefore in order o prove he necessiy of he condiion 2 of Theorem 1 one should check ha ± 0 = D ± 0 0. Sysem 3.1 can be wrien in l 2 as he equaion g ± F ± 2 0g± = 0. As F 2 ± 0 is a kernel operaor, we can apply he Fredholm heory o his equaion, according o which is rivial solvabiliy is equivalen o he condiion ha dee + F 2 ± 0 is no equal o zero [11]. Necessiy of he condiion 2 is proved. Sufficiency: Le us sudy 2.13 in deail. I is known [7] ha K ±, s can be expressed by Ψ ± and soluions P ±, s, Q ±, s of he Marchenko ype equaions 2.13 by he replacemen of Ψ ± by 1 and ±i. Then where K ±, s = Ψ α ±, s + Ψ ± β, s 3.5 α ±, s = 1 2 [P ±, s iq ±, s], 3.6 β, s = 1 2 [P ±, s ± iq ±, s], 3.7 [Ψ ± ] 2 = 1 [α ±, u β ±, u] du 1 [α, u β, u] du, 3.8 from which we uniquely define Ψ ±. We also ake ino accoun ha he sign of Ψ ± is fixed from condiion lim Ψ ± = 1. Thus for furher sudies we should consider he following equaions P ±, s = z ± + s + Q ±, s = ±iz ± + s + P, uz ± u + s du, 3.9 Q, uz ± u + s du. 3.10

8 62 R. EFENDIEV Rewriing 3.9 in he form P ±, s = z ± + s + + z u + z ± u + s du + P ±, τz u + τz ± u + s du dτ in he space l 2 we inroduce he operaor F ± 1 given by he marix F ±1 mn = 2S± n m + n e m+n/2 ; Re > Le s muliply he equaion 3.11 by e mu/2 and inegrae i over s [,. Then we obain p ± = F ± 1 e + F ± 2 e + p± F ± 2, 3.13 in which he operaors F ± 1, F ± 2 are defined by he marix 3.4, 3.12, { e = {e n/2 E} ; p ± = } P ±, ue nu/2 du. As F 2 ± is he race class for 0 and he condiion dee F 2 ± 0 holds, here exiss he inverse operaor R ± = 1 F 2 ± 1 bounded in l 2. Since F 1 ± e, F 2 ± e l 2, hen from 3.13 we ge p ± = R ± [F 1 + F ± 2 ]e Now, denoing f, g = f ng n, we find from 3.11 ha P ±, s = e, B ± s + e, A ± s, + p ±, A ± s, = where = e, B ± s + e, A ± s, + R ± F ± 1 + F ± 2 e, A± s, = = e, B ± s + R ± F ± 1 + F ± 2 + 1e, A± s,, 3.15 B ± s = {B ± ms = S ± me ms/2, s > 0} m=1 and A ± s, = { A ± ms, = 2S m S± k m + k e ks/2 e m+k/2 ; s, > 0 }. Now assume ha he condiions of he heorem are fulfilled. Le us define he funcion P ±, s by he equaliy 3.15 a 0 u according o he given above

9 OPERATOR PENCILS WITH COMPLEX PERIODIC COEFFICIENTS 63 consideraions. Then a u we have P ±, s P ±, τz u + τz ± u + s du dτ = = e, B ± s + R ± F 1 ± + F 2 ± + 1e, A± s, e, B ± τ + R ± F 1 ± + F 2 ± + 1e, A± τ, eτ, A ± s, dτ = = e, B ± s + e, A ± s, + R ± F ± 1 + F ± 2 e, A± s, F ± 1 e, A± s, F ± 2 e, A± s, = e, B ± s + e, A ± s, = z ± + s + For Q ±, s we similarly obain, ha where R ± F ± 2 F ± 1 + F ± 2 e, A± s, = + z u + z ± u + sdu Q ±, s = ±i e, B ± s i e, A ± s, + Q ±, A ± s, So, we have Q ± = ±ir ± [F ± 1 F ± 2 ]e. Lemma 5. For any 0 he kernel K ±, s of he ransformaion operaor and he funcion Ψ ± saisfies he basic equaion K ±, s = Ψ ± z ± + s + K, uz ± u + s du. A unique solvabiliy of he basic equaion follows from Lemma 3. By he direc subsiuion i is easy o calculae ha he soluion of he basic equaion is K ±, u = 1 V ± 2i nαe α e u n/2, Ψ ± = 1 + V n ± e n, α=n where he numbers V ± nα, V ± n V ± mm = S ± m, are defined by he recurren relaions α V m,α+m ± = S m ± Vα Vnα +. n + m Passing o he proof of he basic saemen of he heorem, ha he poenials p and q have form 2.3 le us firs esablish he esimaions for he marix elemens R mn of he operaor R R mn ± δ mn + C 0 Sn, 3.16 d j d j R± mn C j Sn, j = 1, 2, 3.17 where S n = maxs n, S + n and C n, n N, are consans.

10 64 R. EFENDIEV Indeed, from he ideniy R ± = E + R ± F 2 ± i follows ha 1/2 1/2 R mn ± δ mn + R mp ± 2 F pn ±2 2 p=1 p=1 2 1/2 δ mn + 2 R ± R ± 4Sn S ± k pp p + kn + k δ mn + 2 p=1 R ± R ± pp δ mn + R l2 i2 S n. p=1 1 p S ± k 1/2 S n On he oher hand, as i has been noed, he operaor-funcion R ± exiss and is bounded in l 2 because F 2 ± is a kernel operaor a 0 and ± = dee + F 2 ± 0 ha proves he firs inequaliy In order o prove he second esimaion 3.17 we use he ideniy d d R± = R ± F 2 ± R± and by means of he firs esimaion 3.16 obain ha d d R± mn R mq ± ±2 F qp R± pn q=1 p=1 q=1 p=1 δ mn + C 1 Sq Sp δ pn + C 2 Sn C 3 Sp Sn C 4 Sn. p=1 The esimaion d 2 d 2 R± mn C 5 Sn can be proved analogously. Using hese esimaions, from 2.13 and 2.14 one can esablish he correcness of he esimaions d 2 d 2 P ±, s C 6, d 2 d 2 Q±, s C 7. Thus, he funcions K ±, s and Ψ ± have he second derivaives over. From his we conclude ha he series α2 V α ± and n 1 n α=1 α+n V nα ± are convergen. The forms of he coefficiens p and q are direcly deermined from he form of he funcions K ±, s, Ψ ± employing he formulas 2.14, We obain ha for he numbers p n and q n he recurren relaions are correc and hence he series n p n <, q n < converges. Le, finally, {Ŝ± n } be a se of specral daa of he operaor L wih he consruced poenials px, qx Q 2 +. For compleing he proof i remains o show, ha

11 OPERATOR PENCILS WITH COMPLEX PERIODIC COEFFICIENTS 65 {S n ± } coincides wih he iniial se {Ŝ± n }. This follows from he equaliy S n ± = V nn ± = Ŝ± n. The heorem is proved. References [1] T. Akosun and M. Klaus, Inverse heory: problem on he line, Scaering. Vol. 1, 2, Academic Press Inc., San Diego, CA, 2002, Ch , pp MR [2] P. Deif and E. Trubowiz, Inverse scaering on he line, Comm. Pure Appl. Mah , no. 2, MR [3] R. F. Efendiev, An inverse problem for a class of second-order differenial operaors, Dokl. Nas. Akad. Nauk Azerb , no. 4 6, Russian. MR [4] R. F. Efendiev, Specral analysis of a class of nonselfadjoin differenial operaor pencils wih a generalized funcion, Teore. Ma. Fiz , no. 1, Russian. MR English ranslaion: Theore. and Mah. Phys , no. 1, [5] M. G. Gasymov, Specral analysis of a class of second-order nonselfadjoin differenial operaors, Funksional. Anal. i Prilozhen , no. 1, 14 19, 96 Russian. MR English ranslaion: Func. Anal. Appl , no. 1, [6] I. C. Gohberg and M. G. Krein, Inroducion o he heory of linear non-selfadjoin operaors in Hilber space, Izda. Nauka, Moscow, 1965 Russian. MR [7] M. Jaulen and C. Jean, The inverse s-wave scaering problem for a class of poenials depending on energy, Comm. Mah. Phys , MR [8] V. A. Marchenko, Surm Liouville operaors and applicaions, Operaor Theory: Advances and Applicaions, vol. 22, Birkhäuser Verlag, Basel, MR [9] A. Melin, Operaor mehods for inverse scaering on he real line, Comm. Parial Differenial Equaions , no. 7, MR [10] L. A. Pasur and V. A. Tkachenko, An inverse problem for a class of one-dimensional Schrödinger operaors wih complex periodic poenial, Izv. Akad. Nauk SSSR Ser. Ma , no. 6, Russian. MR English ranslaion: Mah. USSR-Izv , no. 3, [11] V. I. Smirnov, A course of higher mahemaics. Vol. IV, Gosudarsv. Izda. Tehn.-Teor. Li., Moscow-Leningrad, 1951 Russian. MR English ranslaion: Pergamon Press, Oxford-New York; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, Insiue of Applied Mahemaics, Baku Sae Universiy Z. Khalilov, 23, AZ1148, Baku, Azerbaijan address: rakibaz@yahoo.com