INVERSE SCATTERING WITH FIXED ENERGY AND AN INVERSE EIGENVALUE PROBLEM ON THE HALF-LINE

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volme 358, Nmber 11, November 6, Pages S -9947(6) Aricle elecronically pblished on Jne 13, 6 INVERSE SCATTERING WITH FIXED ENERGY AND AN INVERSE EIGENVALUE PROBLEM ON THE HALF-LINE MIKLÓS HORVÁTH Absrac. Recenly A. G. Ramm (1999) has shown ha a sbse of phase shifs δ l, l =, 1,..., deermines he poenial if he indices of he known shifs saisfy he Münz condiion l,l L 1 =. Weprovehenecessiyofhis l condiion in some classes of poenials. The problem is redced o an inverse eigenvale problem for he half-line Schrödinger operaors. 1. Inrodcion Consider he inverse scaering problem for he operaor ϕ (r) λ 1/4 (1.1) r ϕ(r) q(r)ϕ(r)+k ϕ(r) =, r, wih fied energy k = 1. Sppose ha he poenial q saisfies rq(r) L 1 (, ). I is known [6] ha for Rλ here eiss a solion of (1.1), niqe p o a consan mliple, saisfying he bondary condiions (1.) ϕ(r, λ) =O(r λ+1/ ), r +, (1.3) ϕ(r, λ) =A(λ)sin(r π/ (λ 1/) + δ(λ)) + o(1), r. The vales δ(λ) C are called phase shifs; hey are defined by (1.3) only mod π. In qanm mechanics mos relevan are he shifs (1.4) δ l = δ(l +1/), l =, 1,,... Concerning he recovery of he poenial q(r) by a se of phase shifs δ(λ n )we menion he following recen resl of Ramm. Theorem 1.1 ([9]). Sppose ha q(r) =for r>aand rq(r) L (,a). Le L N and sppose ha he Münz condiion (1.5) 1 l = l,l L is valid. Then he daa δ l,l L, niqely deermine he poenial q(r). We show ne ha here L can be sbsied by L 1 and ha he condiion (1.5) is almos necessary. Received by he ediors April, 3 and, in revised form, December 1, 4. Mahemaics Sbjec Classificaion. Primary 34A55, 34B; Secondary 34L4, 47A75. Key words and phrases. Inverse scaering, inverse eigenvale problem, m-fncion, compleeness of eponenial sysems. This research was sppored by Hngarian NSF Grans OTKA T 3374 and T c 6 American Mahemaical Sociey License or copyrigh resricions may apply o redisribion; see hp://

2 516 MIKLÓS HORVÁTH Theorem 1.. Le q(r) =for r>aand rq(r) L 1 (,a). Consider arbirary differen comple nmbers λ n wih Rλ n γ>. If (1.6) Rλn λ n =, hen he poenial q(r) is niqely deermined by he shifs δ(λ n ). Theorem 1.3. Le <σ< be arbirarily fied and consider he class B σ = {q(r) :q(r) =for r > a, r 1 σ q(r) L 1 (,a)}. Finally le λ n C be arbirary nmbers wih Rλ n γ>. Nowif (1.7) Rλn λ n <, hen he shifs δ(λ n ) are no enogh o recover he poenial in B σ ;inoherwords, for every q B σ here eiss a differen q B σ sch ha δ(q,λ n )=δ(q, λ n ) n. Remark. If he nmbers λ n are of he form l +1/, hen (1.5) and (1.6) are eqivalen. Ths Theorem 1. eends Theorem 1.1. Remark. The poenials of Theorem 1.1 belong o B σ for every <σ<1/. Hence if (1.5) does no hold, hen here eiss anoher poenial q B σ wih he same daa δ l,l L, asforq. Wheher here eiss a poenial q wih rq (r) L (,a) and δ l (q )=δ l (q), l L, is an open qesion. Analogos L p -problems are no invesigaed here. Ne we consider he inverse eigenvale problem for he Schrödinger operaors (1.8) y +Q()y = λ y,. I is known ha for Q L 1 (, ) he operaor is a limi poin a infiniy and he (essenially niqe) L -solion saisfies he asympoical formla (1.9) y(, λ) =c(λ)e iλ (1 + o(1)), Iλ >; see Theorem.1 below. Consider he bondary condiion (1.1) y() cos α + y () sin α = for some α < π. The vales λ for which he sysem (1.8), (1.1) has a nonrivial L -solion are called eigenvales of he Schrödinger operaor (1.8) wih bondary condiion (1.1). I is known ha he eigenvales are negaive. We apply he noaion λ σ(q, α) for he eigenvales of (1.8), (1.1). We say ha he vales λ n < arecommon eigenvales of he poenials Q and Q if here eis nmbers α n <πwih (1.11) λ n σ(q,α n ) σ(q, α n ) n. In oher words, he bondary condiions can be differen for every eigenvale λ n. Remark. By he above seing every negaive vale λ n < can be considered as an eigenvale of Q if we define α n correspondingly. However Q and λ n define α n, and hence (1.11) conains real informaion, namely ha he parameer α n is hesameforq and Q. This idea is sefl since i is inimaely conneced wih License or copyrigh resricions may apply o redisribion; see hp://

3 INVERSE SCATTERING AND INVERSE EIGENVALUE PROBLEM 5163 he problem of recovering he poenial from phase shifs; see he end of Secion 3 below. Theorem 1.4. Le Q L 1 (, ) and consider he differen nmbers λ n = ik n, inf k n >. If 1 (1.1) =, k n hen he eigenvales λ n deermine Q, i.e. here are no oher poenials Q L 1 (, ) sch ha he λ n are common eigenvales of Q and Q in he above-defined sense. Theorem 1.5. For <δdefine C δ = {Q : Q δ = Q() e δ d < }. Consider he nmbers λ n = ik n, inf k n >. If 1 (1.13) <, k n hen he eigenvales λ n = kn do no deermine he poenial in C δ, i.e. for every Q C δ here eiss Q C δ,q Q sch ha he λ n are common eigenvales of Q and Q. Remark. As we shall check by a Lioville ransformaion, he saemens in Theorems 1.4 and 1.5 are special cases of Theorems 1. and 1.3, respecively. Remark. Condiion (1.6) is necessary and sfficien for he sysem {r λ n } o be closed in L 1 (,a). Analogosly (1.1) holds if and only if {e kn } is closed in L 1 (, ); see Lemma 3. below. The saemens corresponding o Theorems 1.4 and 1.5 for Schrödinger operaors over a finie inerval have been obained earlier by he ahor. Namely, define σ(q, α) as he specrm of he operaor (1.14) y + Q()y = λ y, [,π], (1.15) y() cos α + y () sin α =, y(π) =. We say ha he vales λ n are common eigenvales of he poenials Q and Q if here are nmbers α n <πsaisfying (1.16) λ n σ(q,α n ) σ(q, α n ). The poenial Q L 1 (,π) is said o be deermined by he eigenvales λ n R if here is no oher poenial Q L 1 (,π) saisfying (1.16) for all n. Theorem 1.6 ([5]). Le λ n be arbirary real nmbers saisfying λ n. The poenial Q L 1 (,π) is defined by he eigenvales λ n if and only if he sysem e(λ) = {e ±iµ,e ±iλn : n 1} is closed in L 1 (, π); here µ ±λ n is an arbirary nmber. Le Q L loc 1 (, ) be a poenial which is a limi poin a infiniy, and for Iλ >ley 1 (, λ) be he (essenially niqe) solion of y + Qy = λ y, y L (, ). Define he m-fncion of Q by m(λ) = y 1 (,λ), Iλ >. y 1 (,λ) License or copyrigh resricions may apply o redisribion; see hp://

4 5164 MIKLÓS HORVÁTH As is well known, he m-fncion defines he poenial. Theorem 1.7 (Borg [1], Marchenko [8]). If he m-fncions of Q and Q are he same, m(q,λ) m(q, λ), henq = Q a.e. Using he noion of m-fncions we can generalize Theorems 1.4 and 1.5 a bi frher in he following form. Theorem 1.8. Le λ n be arbirary differen comple nmbers wih Iλ n γ> and sppose ha he m-fncion m(λ) is generaed by a poenial Q L 1 (, ). If (1.17) Iλn λ n =, hen he vales m(λ n ) define he m-fncion (and he poenial). In oher words, if m(q,λ n )=m(q, λ n ) n (allowing ha boh sides be infinie), hen m(q,λ) m(q, λ) and Q = Q a.e. Theorem 1.9. Le δ> and consider differen nmbers λ n, Iλ n γ>. The vales m(λ n ) of he m-fncion generaed by Q C δ defines m(λ) if and only if (1.17) holds. In oher words, if (1.17) is no valid, hen for every Q C δ here eiss a differen Q C δ saisfying m(q,λ n )=m(q, λ n ) n. For relaed resls on finie inervals see [5]. Remark. Theorems 1.8 and 1., respecively 1.9 and 1.3 are idenical; see (3.3) below. In he proofs below we borrow some ideas and mehods from [5]. Secion is devoed o collec he preliminary maerial needed, while Secion 3 conains he proofs of Theorems 1. o Preliminaries We firs recall he following known resl. Theorem.1 (Berezin, Shbin [3]). Le Q L 1 (, ) and λ C \{}. Then he eqaion y + Qy = λ y has wo solions, (.1) y 1 () =e iλ (1 + o(1)), y () =e iλ (1 + o(1)),. If Iλ, hen he solion y 1 reglar a infiniy saisfies he inegral eqaion y 1 () =e iλ sin λ( ) (.) + Q()y 1 () d. λ The fncion y 1 () =y 1 (, λ) is holomorphic in λ C +. Consider wo poenials Q, Q L 1 (, ), and denoe by y1 (, λ), y 1(, λ) he corresponding y 1 -solions. Define he fncions (.3) F (, λ) =y 1(, λ)y1(, λ) y1 (, λ)y 1 (, λ), (.4) F (λ) =F (,λ). By (1.11) he common eigenvales of Q and Q arepreciselyhevales k,k>, where F (ik) =. The analogos fncions in he case of finie inervals are sed e.g. in Geszesy and Simon [4]. License or copyrigh resricions may apply o redisribion; see hp://

5 INVERSE SCATTERING AND INVERSE EIGENVALUE PROBLEM 5165 Lemma.. (.5) F (λ) = Proof. (.) implies (.6) (Q () Q())y 1(, λ)y 1 (, λ) d for Iλ, λ. y 1(, λ) =iλe iλ cos λ( )Q()y 1 (, λ) d, and here he inegral can be esimaed by ( O e Iλ( ) Q() e Iλ) ( d = e Iλ O Conseqenly (for fied λ) (.7) y 1(, λ) =iλe iλ (1 + o(1)),. Comparing his wih (.1) and (.3) gives (.8) Now we have F (, λ),λ fied, Iλ, λ. F (, λ) =y 1 (, λ)y1(, λ) y 1 (, λ)y1 (, λ) =(Q() Q ())y1(, λ)y 1 (, λ), whence F (λ) F (N,λ) = N ) Q. (Q () Q())y 1(, λ)y 1 (, λ) d. Taking he limi N gives (.5). Lemma.3. F (λ) is bonded in every half-plane {λ : Iλ γ>}. Proof. The fncion z(, λ) =y 1 (, λ)e iλ =1+o(1) is bonded for by a bond depending on λ. From e iλ( ) 1 z(, λ) =1+ Q()z(, λ) d iλ we obain z(, λ) 1+ z Q, λ i.e. Q z 1+ z. λ This means ha z if λ Q, or (.9) y 1 (, λ) e Iλ if λ Conseqenly from (.5) ( F (λ) 4 Q Q if λ ma Q, Q, Iλ. ) Q, Iλ. This esimae wih he coniniy of F shows is bondedness on Iλ γ. License or copyrigh resricions may apply o redisribion; see hp://

6 5166 MIKLÓS HORVÁTH From now on we consider more rapidly decaying poenials saisfying (.1) This means ha he fncion belongs o L 1 (, ); indeed, Define σ() d = Q() d <. σ() = σ 1 () = We need he following well-known facs. Q Q() d d = σ. Q() d. Lemma.4 (Marchenko [7], Chaper III). Sppose (.1). Then here eiss a fncion K(, ), coninos for <, saisfying he properies (.11) y 1 (, λ) =e iλ + K(, )e iλ d for, Iλ, (.1) K(, ) =1/ Q, (.13) ( ) + K(, ) 1/σ e σ 1() σ 1 ( + ). Define frher (.14) H(, v) =K( v, + v), v <. Then (.15) H(, v) = H n (, v), where (.16) (.17) Finally we have H (, v) =1/ H n (, v) = v Q, n= Q(α β)h n 1 (α, β) dβ dα, n 1. H n (, v) 1/σ() [σ 1( v) σ 1 (v)] n (.18), n. n! Le Q, Q C δ (as in Theorem 1.5) and define he kernels K(, ), K (, ) corresponding o Lemma.4. Inrodce he hird kernel (.19) K 1 (,, Q, Q )=K(, )+K (, ) + K(, )K (, ) d, <, License or copyrigh resricions may apply o redisribion; see hp://

7 INVERSE SCATTERING AND INVERSE EIGENVALUE PROBLEM 5167 and he corresponding Volerra-ype inegral operaor A Q : C δ C δ, (A Q h)() =h()+ Is relevance is jsified by he following formla. Lemma.5. (.) F (λ) = K 1 (,, Q, Q )h() d. e iλ [A Q (Q Q)] () d. Proof. I is a simple calclaion by sbsiing (.11) ino (.5): [ ] F (λ) = (Q () Q()) e iλ + K(, )e iλ d [ ] e iλ + K (, )e iλ d d. By inerchanging he order of inegraions we ge and = = = = = (Q () Q()) (Q () Q()) (Q () Q()) (Q () Q()) K(, )e iλ(+) d d (Q () Q()) K(, τ )e iλτ dτ d τ e iλτ (Q () Q())K(, τ ) d dτ K(, ) K(, ) e iλ (Q () Q()) + K (, τ)e iλ(+τ) dτ d d K (, )e iλ d d d e iλ K(, )K (, ) d d d K(, )K (, ) d d d. We sed he fac ha K(, ) d is bonded in ; see (.13). Finally we ge F (λ) = + + which is (.). e iλ[ Q () Q()+ (Q () Q())K (, ) d (Q () Q()) (Q () Q())K(, ) d ] K(, )K (, ) d d d, Define he kernel K(,, Q, Q )=e δ( ) K 1 (,, Q, Q ) License or copyrigh resricions may apply o redisribion; see hp://

8 5168 MIKLÓS HORVÁTH and he corresponding Volerra operaor K : L 1 (, ) L 1 (, ), ( Kh)() = K(,, Q, Q )h() d. Lemma.6. Le Q, Q C δ for some δ>. Then a) K : L 1 (, ) L 1 (, ) is compac, and b) A Q : C δ C δ is an (ono) isomorphism. Proof. From σ 1 () = Q(τ) dτ d e δ e δτ Q(τ) dτ d Q δ e δ d = 1 δ e δ Q δ and from (.13) we ge (.1) K(, ) c(d, δ) + Q if Q δ D. Conseqenly [ (.) K 1 (,, Q, Q ) c(d, δ) ( Q + Q ) ( )( ) ] + Q Q d if Q δ, Q δ D, + + and a corresponding bond can be given for K afer he mliplicaion by e δ( ). The operaor norm of K has he bond (.3) as i is seen from K 1,1 sp K(, )h() d d K(,, Q, Q ) d h() K(, ) d d (he dependence of K on Q and Q is no indicaed). In proving a) we will approimae K in operaor norm by operaors of finiedimensional range. Firs le { K(, ) for N, K N (, ) = for >N. We will check ha (.4) Indeed, sp K K N 1,1 if N. K(, ) K N (, ) d =ma(i 1,I ), License or copyrigh resricions may apply o redisribion; see hp://

9 INVERSE SCATTERING AND INVERSE EIGENVALUE PROBLEM 5169 where I 1 = sp N N K(, ) d, I =sp K(, ) d. >N In I 1 we have by (.) (.5) e δ( ) Q(τ) dτ d = N N 1/δe δ Q(τ) e δτ dτ if N and (.6) N τ Q(τ) e δ( ) d dτ N ( )( ) e δ( ) Q Q d d + N + ( ) = Q e δ( ) Q (τ) dτ d d + ma(n, + ) + ( ) τ+ = Q Q (τ) + ma(n+,) ( ) 1/δ Q Q (τ) e δτ dτ e 3/δ e δ/ d + ma(n+,) =1/δ Q(ν) ν 1/δe 3/δ Q(ν) e δ/ /δ e δ Q(ν) e δν ma(n+ ma(n+ ν,) ma(n+ ν,) ma(n, + ) e δ( ) d dτ d,) Q (τ) e δτ dτ d dν e 3/δ Q (τ) e δτ dτ ν Q (τ) e δτ dτ dν. e δ/ d dν This epression is small (niformly in ) ifn is large. Indeed, if ma(n + ν, ) N/, hen he inner inegral is small; if ma(n + ν, ) <N/, hen ν>n/and on his domain he oer inegral is small. This verifies ha I 1 ifn.in I we can apply similar maniplaions (wih insead of N), namely for N ( ) e δ( ) Q d 1/δe δ Q(τ) e δτ dτ 1/δe δn Q δ, ( )( ) e δ( ) Q Q d d Ths (.4) is verified. + + /δ e δ Q(ν) e δν Q (τ) e δτ dτ dν /δ e δn Q δ Q δ. License or copyrigh resricions may apply o redisribion; see hp://

10 517 MIKLÓS HORVÁTH We eend he kernel K N byzeroo[,n] and (for fied N and large M) divide [,N] ino M small sqares of edge lengh N/M. In every small sqare define K N,M o be consan, namely he vale of KN a he lef pper vere. Since K N is coninos on he compac se N, KN K N,M is niformly small for large M, ecep for he small sqares along he diagonal =. Inhese ecepional sqares K N K N,M = K N is bonded (by a bond depending on N). Conseqenly sp N K N (, ) K N,M (, ) d, M. Wih (.4) his means ha K can be approimaed by he K N,M in operaor norm. Since he K N,M have finie-dimensional range, K is compac, so saemen a) is proved. Poin b) is a corollary of a). Indeed, a) implies he compacness of he inegral operaor wih kernel K 1. We know A Q = I + K 1. If A Q is no an isomorphism, hen 1 ms be an eigenvale of K 1, i.e. here eiss h C δ sch ha h = K 1 h. B his is impossible for a Volerra operaor wih a coninos kernel. So A Q is an isomorphism as assered. Lemma.7. Le Q, Q,Q C δ for some δ> and sppose ha Q δ, Q δ, Q δ D. Then (.7) (A Q A Q )h δ c(d, δ) Q Q δ h δ, h C δ. The consan c(d, δ) depends only on is argmens. Proof. Consider he fncions H n (, v) defined in Lemma.4. Denoe ϱ(, v) =σ 1 ( v) σ 1 (v), v ; i is increasing in v for fied and decreasing in for fied v since ϱ = v Q <. Analogosly define ϱ wih Q insead of Q. Finally le σ() = Q Q, σ 1 () = The proof of (.7) is based on he esimae (.8) σ, ϱ(, v) = σ 1 ( v) σ 1 (). H n(, v) H n (, v) 1/σ () ϱ(, v) [ϱ(, v)+ϱ (, v)] n 1 (n 1)! ϱ(, v)n +1/ σ() ; n! for n = only he second smmand is considered on he righ. We apply indcion on n. Forn = H (, v) H (, v) =1/ (Q Q) 1/ σ(). License or copyrigh resricions may apply o redisribion; see hp://

11 INVERSE SCATTERING AND INVERSE EIGENVALUE PROBLEM 5171 Sppose (.8) for a vale of n. Then (.9) Hn+1(, v) H n+1 (, v) = + v v We se he ideniy (.3) (Q (α β) Q(α β)) H n(α, β) dβ dα Q(α β)(h n(α, β) H n (α, β)) dβ dα def = I 1 + I. v = Q (α β) Q(α β) dβ dα ( σ(α v) σ(α)) dα = ϱ(, v) and (.18) o obain v I 1 1/ Q (α β) Q(α β) σ (α) ϱ (α, β) n (.31) dβ dα n! 1/σ () ϱ (, v) n v Q (α β) Q(α β) dβ dα n! =1/σ () ϱ(, v) ϱ (, v) n. n! On he oher hand v ϱ(α, β)n (.3) I 1/ Q(α β) σ(α) dβ dα n! v +1/ Q(α β) σ (α) ϱ(α, β) [ϱ(α, β)+ϱ (α, β)] n 1 dβ dα (n 1)! def = I 1 + I. Apply he ideniy o ge (.33) (.34) v ϱ(α, v) n I 1 1/ σ() n! =1/ σ() Q(α β) dβ = ϱ (α, v) α ϱ(, v)n+1, (n +1)! n 1 I 1/σ () ϱ(, v) k= v Q(α β) dβ dα ( n 1 ) k (n 1)! ϱ (, v) n 1 k v ϱ(α, v) k Q(α β) dβ dα n 1 ( n 1 ) =1/σ k 1 () ϱ(, v) (n 1)! k +1 ϱ (, v) n 1 k ϱ(, v) k+1 k= ( n n =1/σ () ϱ(, v) k) n! ϱ (, v) n k ϱ(, v) k. k=1 License or copyrigh resricions may apply o redisribion; see hp://

12 517 MIKLÓS HORVÁTH Smming p (.31)-(.34) gives H n+1(, v) H n+1 (, v) 1/σ () ϱ(, v) [ϱ(, v)+ϱ (, v)] n which verifies (.8) for n +1. Conseqenly ϱ(, v)n+1 +1/ σ() (n +1)! n! (.35) H (, v) H(, v) 1/σ () ϱ(, v)e ϱ(,v)+ϱ (,v) +1/ σ()e ϱ(,v). Since we have ϱ(, v) = σ 1 ( v) σ 1 () σ 1 () = Q () Q() d c(δ) Q Q δ, he esimae (.35) can be conined as follows: [ (.36) H (, v) H(, v) c(d, δ) Q Q + Q Q δ or, from (.14) ] Q (.37) K (, ) K(, ) [ c(d, δ) Q Q + Q Q δ + + ] Q. Now we are able o prove he esimae (.38) e δ K1 (,, Q, Q ) K 1 (,, Q, Q ) d C(D, δ) Q Q δ for all. This implies he Lipschiz propery (.7) becase A Q A Q sp Now se he decomposiion (.19) in e δ( ) K 1 (,, Q, Q ) K 1 (,, Q, Q ) d. (.39) + def = I 1 +I. e δ K 1 (,, Q, Q ) K 1 (,, Q, Q ) d e δ K (, ) K (, ) d e δ K(, ) K (, ) K (, ) d d License or copyrigh resricions may apply o redisribion; see hp://

13 INVERSE SCATTERING AND INVERSE EIGENVALUE PROBLEM 5173 We esimae I 1 sing (.37) as follows: e δ Q (τ) Q (τ) dτ d and analogosly hence = 1/δ Q (τ) Q (τ) τ e δ d dτ Q (τ) Q (τ) e δτ dτ =1/δ Q Q δ ( Q Q δ e δ ) Q d D/δ Q Q δ, I 1 c(d, δ) Q Q δ. In I we have, as in verifying (.4) above, ( ) e δ K(, ) Q Q d d + ( )( ) c(d, δ) e δ Q Q Q d d c(d, δ) + + Q(ν) e δν Q (ξ) Q(ξ) e δξ dξ dν c(d, δ) Q Q δ, and by he same way we obain ( Q Q δ e δ K(, ) c(d, δ) Q Q δ. + ) Q d d Ths we have checked he esimae (.38). The proof of Lemma.7 is complee. 3. Proof of he main resls Proof of Theorem 1.8. Sppose indirecly ha (1.17) holds, however here are wo poenials Q,Q L 1 (, ) wihm(q,λ n )=m(q, λ n ), n. Inrodce he fncion F (λ) as in (.4); hen F (λ n )=, n. Leµ> be arbirary; by Lemma.3 he fncion F (λ + iµ) is bonded analyic in he pper half-plane Iλ >. So if F is no idenically vanishing, is zeros ms saisfy he Blaschke condiion Iλ 1+ λ < ; F (λ+iµ)=,iλ> see e.g. Dren []. In pariclar, for he zeros λ n iµ Iλ n µ < for all µ>. 1+ λ n iµ Iλ n >µ License or copyrigh resricions may apply o redisribion; see hp://

14 5174 MIKLÓS HORVÁTH We can sppose ha γ>µ and hen Iλn λ n c(µ) Iλ n µ 1+ λ n iµ <, a conradicion. Conseqenly F (λ), i.e. m m, and by Theorem 1.7, Q = Q a.e. In order o prove Theorem 1.9 we need Lemma 3.1 ([5]). Le B 1 and B be Banach spaces. For every q B 1 le a coninos linear operaor be defined so ha for some q B 1 A q : B 1 B (3.1) A q : B 1 B is an (ono) isomorphism and he mapping q A q is Lipschizian in he sense ha (3.) (A q A q )h c(q ) q q 1 h 1 for all h, q, q B 1 wih q 1, q 1 q 1, he consan c(q ) being independen of q, q and h. Then he se {A q (q q ):q B 1 } conains a ball in B wih cener a he origin. The Münz heorem abo he closedness of eponenial sysems is known in several versions. The ahor did no find a proper reference conaining he version formlaed below, so a proof is provided. Lemma 3.. Le λ n be differen comple nmbers wih Iλ n, lim inf Iλ n >. Then (3.3) { e iλ n } is closed in L 1 (, ) Iλ n 1+ λ n =. Proof. The if par: Le h L 1 (, ), h()e iλn d =foralln. Define H(z) = h()e iz d, Iz. I is bonded analyic in he pper half-plane, so if H, is zeros saisfy he Blaschke condiion and hen Iλ n > Iλ n 1+ λ n <, a conradicion. This implies H andhenh =, i.e. he sysem {e iλn } is closed in L 1. The only if par: Le Iλ n 1+ λ n <. Iλ n > Since lim inf λ n >, here eiss δ> sch ha Iλ n δ wih finiely many ecepions. Ths (3.4) I(λn + iδ) 1+ λ n + iδ <. License or copyrigh resricions may apply o redisribion; see hp://

15 INVERSE SCATTERING AND INVERSE EIGENVALUE PROBLEM 5175 This means ha here eiss a Blaschke prodc B(z) wihb(λ n + iδ) = n; see []. The fncion B(z) z+i belongs o he Hardy space H (C + ). By he Paley-Wiener heorem here eiss g L (, ) wih B(z) z + i = g()e iz d. Conseqenly wih = g()e i(λ n+iδ) d = Ths {e iλ n } is no closed in L 1. g()e δ e iλ n d = h() =g()e δ L 1 (, ). h()e iλ n d Proof of Theorem 1.9. Le Q C δ be arbirary; or ask is o find a differen Q C δ wih m(q,λ n )=m(q, λ n ), i.e. F (λ n ) =. Sppose ha (1.17) is no re. Then (3.5) I(λn + iδ) λ n + iδ < (becase Iλ n γ>). By Lemma 3. here eiss g L 1 (, ) wih (3.6) = g()e (iλ n δ) d = (3.7) h() =g()e δ C δ. Recall from Lemma.5 he ideniy F (λ) = h()e iλ n d, e iλ [A Q (Q Q)] () d. Comparing his wih (3.6) we see ha F (λ n ) = is garaneed if (3.8) A Q (Q Q) =ch for some c. To check (3.8) we apply Lemma 3.1 wih B 1 = B = C δ. The condiions (3.1) and (3.) are verified in Lemmas.6 and.7, respecively. Since he se {A Q (Q Q) : Q C δ } conains all elemens of C δ of sfficienly small norm, here eiss a poenial Q C δ,q Q saisfying (3.8). Now from F (λ n ) = i follows ha m (λ n )=m(λ n ), hogh m m. Proof of Theorems 1.4 and 1.5. If λ n = ik n, inf k n = γ >, hen m (λ n ) = m(λ n ) if and only if he vales λ n = kn are common eigenvales of Q and Q. Since Iλ n / λ n =1/k n, Theorems 1.4 and 1.5 are conained in Theorems 1.8 and 1.9, respecively. Proof of Theorems 1. and 1.3. Le Q L 1 (, ) and consider he solion y 1 (, λ) = e iλ (1 + o(1)) of y + Qy = λ y wih Iλ. Inrodce he fncion (3.9) ϕ(r, λ) = ry 1 (ln a r,iλ) for <r a, Rλ. A shor calclaion gives ha (3.1) ϕ (r, λ) λ 1/4 r ϕ(r, λ) 1/r Q(ln a r )ϕ(r, λ) =, <r a. License or copyrigh resricions may apply o redisribion; see hp://

16 5176 MIKLÓS HORVÁTH This means ha ϕ saisfies (1.1) (for r a) if (3.11) q(r) =1/r Q(ln a r )+1. We observe ha (3.1) rq(r) L 1 (,a) Q() L 1 (, ). Indeed, sbsiing =ln a r gives a 1 a Q() d = r Q(ln a r ) dr = r q(r) 1 dr, and his is finie if and only if rq(r) L 1 (,a). Analogosly for <δ< (3.13) r 1 δ q(r) L 1 (,a) Q C δ since a Q() e δ d = a δ r 1 δ q(r) 1 dr. From y 1 (, λ) =e iλ (1 + o(1)) we infer (3.14) ϕ(r, λ) =a λ r λ+1/ (1 + o(1)), r +, which corresponds o (1.). In he poenial-free domain r a (1.1) has he form (3.15) ϕ + ϕ = λ 1/4 r ϕ, r a. Is general solion can be given by Bessel fncions (3.16) ϕ(r, λ) =c(λ) r [cos α(λ) J λ (r)+sinα(λ) Y λ (r)], wih α(λ) C; see Wason [1]. Taking ino accon he asympoic forms (3.17) J ν (r) = πr cos(r νπ/ π/4) + O(r 3/ ), r, (3.18) Y ν (r) = πr sin(r νπ/ π/4) + O(r 3/ ), r, we ge ϕ(r, λ) =c(λ) cos(r λπ/ π/4 α(λ)) + O(1/r) π = c(λ) sin(r π/(λ 1/) α(λ)) + O(1/r), π which is compaible wih (1.3) if and only if (3.) On he oher hand, δ(λ) α(λ) (modπ). (3.1) ϕ d( (a, λ) cos α(λ) rjλ (r)) dr (a)+sinα(λ) d( ry λ (r)) dr (a) =. ϕ(a, λ) a [cos α(λ)jλ (a)+sinα(λ)y λ (a)] Finally from (3.9) i follows ha (3.) ϕ (a, λ) ϕ(a, λ) = 1/a 1/ y 1 (,iλ) a 1/ y 1 (,iλ). a 1/ y 1 (,iλ) License or copyrigh resricions may apply o redisribion; see hp://

17 INVERSE SCATTERING AND INVERSE EIGENVALUE PROBLEM 5177 Now (3.)-(3.) show ha he knowledge of he shif δ(λ) (forafiedλ, Rλ ) means he knowledge of ϕ (a,λ) ϕ(a,λ) and of y 1 (,iλ) y 1 (,iλ).inoherwordsforq, Q L 1 (, ) and for q, q defined by (3.11), we have (3.3) δ(q,λ) δ(q, λ)(mod π) m(q,iλ)=m(q, iλ) for he corresponding m-fncions. Taking ino accon (3.1) and (3.13) we see ha Theorem 1. and 1.8, respecively Theorem 1.3 and 1.9, are he same saemens. The proof is complee. References 1. G. Borg, Uniqeness heorems in he specral heory of y +(λ q())y =, Proc. 11- h Scandinavian Congress of Mahemaicians (Oslo), Johan Grnd Tanms Forlag, 195, pp MR5863 (15:315a). P. L. Dren, The heory of H p spaces, Academic Press, New York, 197. MR68655 (4:355) 3. M. A. Shbin and F. A. Berezin, The Schrödinger eqaion, Klwer, Dordrech, MR (93i:811) 4. F. Geszesy and B. Simon, Inverse specral analysis wih parial informaion on he poenial, II. The case of discree specrm, Trans. Amer. Mah. Soc. 35 (), MR (j:3419) 5. M. Horváh, Inverse specral problems and closed eponenial sysems, Annals of Mah. 16 (5), MR B. M. Levian, Inverse Srm-Lioville problems, V.N.U. Science Press, Urech, MR93388 (89b:341) 7. V. A. Marchenko, Srm-Lioville operaors and heir applicaions (in Rssian), Naokova Dmka, Kiev, MR (58:1317) 8. V. A. Marchenko, Some qesions in he heory of one-dimensional linear differenial operaors of he second order I. (in Rssian), Trdy Moskov. Ma. Obsc. 1 (195), MR5864 (15:315b) 9. A. G. Ramm, An inverse scaering problem wih par of he fied-energy phase shifs, Comm. Mah. Phys. 7 (1999), MR (g:34136) 1. G. N. Wason, A reaise on he heory of Bessel fncions, Cambridge Universiy Press, Cambridge, MR (96i:331) (review of 1995 ediion) Deparmen for Mahemaical Analysis, Insie of Mahemaics, Technical Universiy of Bdapes, H 1111 Bdapes, Műegyeem rkp. 3-9, Hngary address: horvah@mah.bme.h License or copyrigh resricions may apply o redisribion; see hp://