Effectivized Holder-logarithmic stability estimates for the Gel fand inverse problem

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1 Effectivized Holder-logarithmic stability estimates for the Gel fand inverse problem Mikhail Isaev, Roman Novikov To cite this version: Mikhail Isaev, Roman Novikov. Effectivized Holder-logarithmic stability estimates for the Gel fand inverse problem. Inverse Problems, IOP Publishing, 014, 0 9), pp.19. <hal > HAL Id: hal Submitted on 1 Apr 014 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Effectivized Hölder-logarithmic stability estimates for the Gel fand inverse problem M.I. Isaev and R.G. Novikov Abstract We give effectivized Hölder-logarithmic energy and regularity dependent stability estimates for the Gel fand inverse boundary value problem in dimension d =. This effectivization includes explicit dependance of the estimates on coefficient norms and related parameters. Our new estimates are given in L and L norms for the coefficient difference and related stability efficiently increases with increasing energy and/or coefficient difference regularity. Comparisons with preceeding results are given. 1 Introduction and main results We consider the equation where ψ +vx)ψ = Eψ, x R, 1.1) is an open bounded domain in R, C, 1.) v L ). 1.) Equation 1.1) can be regarded as the stationary Schrödinger equation of quantum mechanics at fixed energy E. Equation 1.1) at fixed E arises also in acoustics and electrodynamics. As in Section 5 of Gel fand s work [9] we consider an operator establishing a relationship between ψ and ψ/ ν on for all sufficiently regular solutions ψ of equation 1.1) in = at fixed E, where ν is the outward normal to. As in [6], [16] for example) we represent such an operator as the irichlet-to-neumann map ˆΦE) defined by the relation where we assume also that ˆΦE)ψ ) = ψ ν, 1.4) E is not a irichlet eigenvalue for operator +v in. 1.5) The map ˆΦ = ˆΦE) can be regarded as all possible boundary measurements for the physical model described by equation 1.1) at fixed energy E under assumption 1.5). We consider the following inverse boundary value problem for equation 1.1): Problem 1.1. Given ˆΦ for some fixed E, find v. 1

3 This problem is known as the Gel fand inverse boundary value problem for the Schrödinger equation at fixed energy E in three dimensions see [9], [6]). For E = 0 this problem can be regarded also as a generalization of the Calderón problem of the electrical impedance tomography in three dimensions see [5], [6]). Problem 1.1 can be also considered as an example of ill-posed problem; see [4], [] for an introduction to this theory. Let, for real m 0, H m R ) = { w L R ) : F 1 1+ ξ ) m Fw L R ) }, w H m R ) = F 1 1+ ξ ) m Fw L R ), 1.6) where F denote the Fourier transform Fwξ) = 1 π) e iξx wx)dx, ξ R. R In addition, for real m 0, we consider the spaces W m R ) defined by W m R ) = { w L 1 R ) : 1+ ξ ) m Fw L R ) }, w W m R ) = 1+ ξ ) m Fw L R ). 1.7) We note that for integer m the space W m R ) contains the standard Sobolev space W m,1 R ) of m-times smooth functions in L 1 on R. In the present work we obtain, in particular, the following theorems: Theorem 1.1. Suppose that satisfies 1.) and v 1,v satisfy 1.), 1.5) for some real E. Suppose also that: v j L ) N for some N > 0, j = 1,; suppv v 1 ), v v 1 H m R ), v v 1 H m R ) N H m for some m > 0 and N H m > 0. Let δ = ˆΦ E) ˆΦ 1 E) L ) L ), 1.8) where ˆΦ 1 E), ˆΦ E) denote the irichlet-to-neumann maps for v 1, v, respectively. Then, there exist some positive constants A,B,α,β depending on only such that v v 1 L ) A αe +β1 τ) ln +δ 1)) )1 δ τ + +B1+N) 4m N H m αe +β1 τ) ln +δ 1)) ) m 1.9) for any τ 0,1] and E 0. Besides, estimate 1.9) is also fulfilled for any τ 0,1) and E < 0 under the following additional condition: αe +β1 τ) ln +δ 1)) > ) Theorem 1.. Suppose that satisfies 1.) and v 1,v satisfy 1.), 1.5) for some real E. Suppose also that: v j L ) N for some N > 0, j = 1,; suppv v 1 ), v v 1 W m R ), v v 1 Wm R ) N W m for some

4 m > and N W m > 0. Let δ be defined by 1.8). Then, there exist some positive constants Ã, B, α, β depending on only such that v v 1 L ) Ã αe + β1 τ) ln +δ 1)) ) 1 δ τ + 1+N)m ) N W m + B αe + m β1 τ) ln +δ 1)) ) m ) for any τ 0,1] and E 0. Besides, estimate 1.11) is also fulfilled for any τ 0,1) and E < 0 under the following additional condition αe + β1 τ) ln +δ 1)) > ) Theorems 1.1 and 1. are proved in Sections and 4, respectively. These proofs are based on Lemmas.1,. and. given in Section. Then these proofs are based on the intermediate estimates.7), 4.8) which may be of independent interest. Remark 1.1. The estimates of Theorem 1. can be regarded as a significant effectivization of the following estimates of [16] for the three-dimensional case: for E R; v v 1 L ) C 1 N m,,m,e) ln +δ 1)) s 1 1.1) v v 1 L ) C N m,,m,τ)1+ E)δ τ + +C N m,,m,τ)1+ E) s s1 ln +δ 1)) s 1.14) for E 0, τ 0,1) and any s [0,s 1 ]. Here δ is defined by 1.8) and s 1 = m )/. In addition, estimates 1.1) and 1.14) were obtained in [16] under the assumptions that: satisfies 1.), v j satisfies 1.), 1.5), suppv j, v j W m,1 R ), v j W m,1 R ) N m, j = 1,, for some integer m > and N m > 0. Actually, Theorem 1. was obtained in the framework of finding the dependance of C 1, C, C of 1.1), 1.14) on N m, m and τ. One can see that the estimates of Theorem 1. depend explicitely on coefficient norms N, N W m and parameteres m, τ and imply 1.1), 1.14) with some C 1, C, C explicitely dependent on N m, m, τ as a corollary. Besides, in Theorem 1. we do not assume that each of potentials v 1, v is m-times differentiable and is supported in in a similar way with Theorem.1 of [4]). By the way we would like to note also that even for E = 0 the reduction of Hölder-logarithmic stability estimates like 1.9), 1.11) to pure logarithmic estimates like 1.1) is not optimal for large m because of the following asymptotic formula: sup δ τ µ ) µe ln+δ 1 )) µ = O τ) µ as µ +. τ δ 0,1] In particular, even for E = 0 the Hölder-logarithmic estimates 1.9), 1.11) are much more informative than their possible pure logarithmic reductions.

5 Remark 1.. Theorem 1.1 was obtained as an extention of Theorem 1. to the L -norm case. In addition, it is important to note that the second logarithmic ) term of the right-hand side of 1.9) is considerably better than the analogous term of 1.11). In particular, ln R = O +δ 1 )) ) m for δ 0, R = O ) E m for E +, whereas R = O ln +δ 1 )) ) m R = O ) E m 6 for δ 0, for E +, where R and R denote the second logarithmic ) terms of the right-hand sides of 1.9) and 1.11), respectively. Remark 1.. The estimates of Theorem 1.1 should be compared also with the following estimate of [1] for the three-dimensional case: v v 1 H m R ) C E δ + ) ) m ) E +lnδ 1, 1.15) where C = CN m,,suppv v 1 ),m) > 0, v j H m ) N m j = 1,), suppv v 1 ), m > /, δ is the distance between the boundary measurements Cauchy data) for v 1, v and is, roughly speaking, similar to δ of 1.8) and where δ 1/e. A principal advantage of 1.9) in comparison with 1.15) consists in estimation v v 1 in the L -norm instead of the H m -norm. Besides, estimate 1.9) depends explicitely on coefficient norms N, N W m and parameteres m, τ in contrast with 1.15). In addition, in 1.9) we do not assume that each of v 1, v belongs to H m. Remark 1.4. In the literature on Problem 1.1 estimates of the form 1.1) are known as global logarithmic stability estimates. The history of these estimates goes back to [1] for the case when s 1 1 and to [] for the case when s 1 > 1. In addition, estimates of the form 1.9), 1.11), 1.14), 1.15) are known in the literature as Hölder-logarithmic energy and regularity dependent stability estimates. For the case when τ = 1 in 1.9), 1.11) or when s = 0 in 1.14) the history of such estimates in dimension d = goes back to [9], [1], where such energy and regularity dependent rapidly convergent approximate stability estimates were given for the inverse scattering problem. Then for Problem 1.1 energy dependent stability estimates changing from logarithmic type to Hölder type for high energies were given in [0]. However, this high energy stability increasing of [0] is slow. The studies of [9], [1], [], [0] were continued, in particular, in [5], [16], [1] and in the present work. Remark 1.5. In Theorems 1.1, 1. we consider the three-dimensional case for simplicity only. Similar results hold in dimension d >. As regards to logarithmic and Hölder-logarithmic stability estimates for Problem1.1 in dimension d =, we refer to [5], [7], [8]. In addition, for problems like Problem 1.1 the history of energy and regularity dependent rapidly convergent approximate stability estimates in dimension d = goes back to [8]. 4

6 Remark 1.6. In a similar way with results of [17], [18] and subsequent studies of [6], estimates 1.9), 1.11) can be extended to the case when we do not assume that condition 1.5) is fulfiled and consider an appropriate impedance boundary maprobin-to-robin map) instead of the irichlet-to-neumann map. Remark 1.7. Apparently, estimates analogous to estimates of Theorems 1.1 and 1. hold if we replace the difference of tn maps by the difference of corresponding near field scattering data in a similar way with results of[10],[14],[19]. Remark 1.8. The optimalityin different senses) of estimates like1.1), 1.14) was proved in [4], [1], [1]. See also [6], [15] and references therein for the case of inverse scattering problems. Remark 1.9. Estimates 1.9), 1.11) for τ = 1 are roughly speaking coherent with stability properties of the approximate monochromatic inverse scattering reconstruction of [9], [1], implemented numerically in []. Estimates 1.9), 1.11) for E = 0 are roughly speaking coherent with stability properties of the reconstruction of []. In addition, estimates 1.9), 1.11) can be used for the convergence rate analysis for iterative regularized reconstructions for Problem 1.1 in the framework of an effectivization of the approach of [10] for monochromatic inverse scattering problems. Lemmas Let ˆv denote the Fourier transform of v: ˆvξ) = Fvξ) = 1 π) e iξx vx)dx, ξ R..1) R Lemma.1. Suppose that satisfies 1.) and v 1,v satisfy 1.), 1.5) for some real E. Suppose also that v j L ) N, j = 1,, for some N > 0. Let δ be defined by 1.8). Then ) for any ρ > 0 such that ˆv ξ) ˆv 1 ξ) c 1 1+N) e ρl δ + v 1 v L ) E +ρ ξ E +ρ, E +ρ 1+N) r 1, where L = max x x and constants c 1,r 1 > 0 depend on only..) Some version of estimate.) was given in [16] see formula 4.1) of [16]). Lemma.1 is proved in Section 6. This proof is based on results presented in Section 5. Lemma.. Let w H m R ), w H m R ) N H m for some real m > 0 and N H m > 0, where the space H m R ) is defined in 1.6). Then, for any r > 0, 1/ Fwξ) dξ c N H m r m,.) ξ r 5

7 where Fw is defined according to.1) and c = π) /. Proof of Lemma.. Note that Fwξ) dξ 1+ ξ ) m Fw r m ξ r Using 1.6),.4) and the Parseval theorem L R )..4) F w L R ) = π) / w L R ).5) for w F 1 1+ ξ ) m Fw, we get estimate.). Lemma.. Let w W m R ), w Wm R ) N W m for some real m > and N W m > 0, where the space W m R ) is defined in 1.7). Then, for any r > 0, N W m Fwξ) dξ c m r m,.6) ξ r where Fw is defined according to.1) and c = 4π. Proof of Lemma.. Note that r m Fwξ) 1+ ξ ) m/ Fwξ) N W m for ξ r..7) Using.7), we obtain that ξ r Fwξ) dξ + r N W m 4πN t m 4πt Wm dt m r m..8) Proof of Theorem 1.1 Using the Parseval formula.5), we get that v v 1 L ) = π) / ˆv ˆv 1 L R ) π) / I 1 r)+i r)),.1) for r > 0, where ˆv j is defined according to.1) with v j 0 on R \, j = 1,, I 1 r) = ξ r I r) = ξ r ˆv ξ) ˆv 1 ξ) dξ ˆv ξ) ˆv 1 ξ) dξ 1/ 1/,. 6

8 Let r = q1+n) 4/ E +ρ ) 1/, q = 1 π where c 1 is the constant of Lemma.1. Then, using Lemma.1 for ξ r, we get that I 1 r) 4πr c 11+N) 4 e ρl δ + v 1 v L ) E +ρ π) / E +ρ e ρl δ ) 16πc 1/ 1,.) ) 1/ + v 1 v L ) for q1+n) 4/ E +ρ ) 1/ E +ρ and E +ρ 1+N) r 1. In addition, using.), we have that Let r = r ) r 1 be such that ).) I r) c N H m r m..4) E +ρ r = qe +ρ ) 1/ E +ρ..5) Using.1),.).5) with r defined in.), we obtain that E +ρ e ρl δ v v 1 L ) + v 1 v L ) + +π) / 1+N) 4m c q m N H me +ρ ) m,.6) 1 E v +ρ e ρl δ v 1 L ) N)4m q m N H me +ρ ) m,.7) for E +ρ 1+N) r, where L,c are the constants of Lemmas.1,. and q,r are the constants of formulas.),.5). Let τ 0,1) and ue to.7), for δ such that γ = 1 τ L, ρ = γln +δ 1)..8) E + γln+δ 1 ) ) 1+N) r,.9) the following estimate holds: 1 v 1 v L ) 1 E + γln +δ 1)) ) 1/ +δ 1 ) γl δ+ + 1+N)4m q m N H m E + γln +δ 1)) ) m,.10) 7

9 where γ is defined in.8). Note that +δ 1 ) γl δ = 1+δ) 1 τ δ τ 4δ τ for δ 1..11) Combining.10),.11), we get that v v 1 L ) A 1 λ E +γ ln +δ 1)) )) 1 δ τ + +B 1 1+N) 4m N H m λ E +γ ln +δ 1)) )) m.1) for δ 1 satisfying.9) and some positive constants A 1,B 1,λ depending on only. In view of definition 1.6), we have that v v 1 L ) v v 1 H m R ) N H m. Hence, we get that, for 0 < E + γln+δ 1 ) ) 1+N) r, v v 1 L ) 1+N) 4m N H m E +γ ln +δ 1)) r ) m..1) On other hand, in the case when E + γln+δ 1 ) ) 1+N) r and δ > 1 we have that v v 1 L ) c v v 1 L ) c N c E +γ ln +δ 1)) r )1 δ τ,.14) where c = 1dx 1/..15) Combining.8),.1).14), we obtain estimate 1.9). This completes the proof of Theorem Proof of Theorem 1. ue to the inverse Fourier transform formula vx) = e iξxˆvξ)dξ, x R, 4.1) R we have that v 1 v L ) sup e iξx ˆv ξ) ˆv 1 ξ))dξ x Ĩ1r)+Ĩr) 4.) R 8

10 for r > 0, where Ĩ 1 r) = ˆv ξ) ˆv 1 ξ) dξ, Let Ĩ r) = ξ r ξ r ˆv ξ) ˆv 1 ξ) dξ. ) 1/ r = q1+n) / E +ρ ) 1/6 8πc1 c, q =, 4.) where c 1 is the constant of Lemma.1 and c is defined by.15). Then, combining the definition of Ĩ1, Lemma.1 for ξ r and the inequality v v 1 L ) c v v 1 L ), we get that Ĩ 1 r) 4πr c 11+N) e ρl δ + c v 1 v L ) E +ρ 1 E +ρ e ρl δ + v 1 v L ) c for q1+n) / E +ρ ) 1/6 E +ρ and E +ρ 1+N) r 1. In addition, using.6), we get that Let r = r ) r 1 be such that ) 4.4) Ĩ r) c N W m m r m. 4.5) E +ρ r = qe +ρ ) 1/6 E +ρ. 4.6) Using 4.), 4.4) 4.6) with r defined in 4.), we obtain that v v 1 L ) 1 E +ρ e ρl δ + v 1 v L ) + c 1+N) m ) + c m ) q m N W me +ρ ) m 6, 4.7) 1 v v 1 L ) 1 E +ρ e ρl δ+ c 1+N)m ) +4π m ) q m N W me +ρ ) m 6 4.8) for E +ρ 1+N) r, where L, c are the constants of Lemmas.1,. and c, q, r are the constants of formulas.15), 4.), 4.6). Let τ 0,1) and γ = 1 τ L, ρ = γln +δ 1). 4.9) 9

11 ue to 4.8), for δ such that the following estimate holds: 1 v 1 v L ) E + γln+δ 1 ) ) 1+N) r, 4.10) 1 c E + γln +δ 1)) ) 1/ +δ 1 ) γl δ+ 1+N)m ) +4π m ) q m N W m E + γln +δ 1)) ) m 6, 4.11) where γ is defined in 4.9). Note that +δ 1 ) γl δ = 1+δ) 1 τ δ τ 4δ τ for δ ) Combining 4.11), 4.1), we get that v v 1 L ) Ã1 λ E +γ ln +δ 1)) )) 1 δ τ + 1+N) m ) N W m + B 1 λe +γ ln +δ 1)) )) m 6 m 4.1) for δ 1 satisfying 4.10) and some positive constants Ã1, B 1, λ depending on only. Using 1.7) and 4.), we get that ) v v 1 L ) 1+ ξ ) m/ v v 1 W m R ) dξ R N W m + for some c 4 > 0. Here we used also that + 0 4πt 1+t ) m/dt πt dt+ 4πt 1+t ) m/dt c e m 4 m N W m ) 4πt t m dt c ) e m c 4 m m. Using 4.14), we get that, for 0 < E + γln+δ 1 ) ) 1+N) r, v v 1 L ) c 4 1+N) m ) N W m m E +γ ln +δ 1)) e 6 r ) m ) On other hand, in the case when E + γlnδ 1) 1 + N) r and δ > 1 we have that E +γ ln +δ 1)) )1 v v 1 L ) N δ τ. 4.16) Combining 4.9), 4.1), 4.15) and 4.16), we obtain estimate 1.11). This completes the proof of Theorem 1.. r 10

12 5 Faddeev functions Suppose that v L ), v 0 on R \, 5.1) where satisfies 1.). More generally, one can assume that v is a sufficiently regular function on R with sufficient decay at infinity. 5.) Under assumptions 5.), we consider the functions ψ, µ, h: where x R, k C, Imk 0, ψx,k) = e ikx µx,k), 5.) µx,k) = 1+ gx y,k)vy)µy,k)dy, R hk,l) = π) gx,k) = π) R e iξx dξ ξ +kξ, 5.4) R e ik l)x vx)µx,k)dx, 5.5) where k,l C, k = l, Imk = Iml 0. Here, 5.4) at fixed k is considered as a linear integral equation for µ, where µ is sought in L R ). The functions ψ, h and G = e ikx g are known as the Faddeev functions, see[7], [8], [11],[6]. Thesefunctions wereintroduced forthe first time in [7], [8]. In particular, we have that +k )Gx,k) = δx), +vx))ψx,k) = k ψx,k), where x R, k C \R. We recall also that the Faddeev functions G, ψ, h are some extension to the complex domain of functions of the classical scattering theory for the Schrödinger equation in particular, h is an extension of the classical scattering amplitude). Note also that G, ψ, h in their zero energy restriction, that is for k = 0, l = 0, were considered for the first time in []. The Faddeev functions G, ψ, h were, actually, rediscovered in []. For further considerations we will use the following notations: Σ E = { k C : k = k 1 +k +k = E }, Under assumptions 5.), we have that: Θ E = {k Σ E, l Σ E : Imk = Iml}, k = Rek + Imk ) 1/ for k C. µx,k) 1 as k, 5.6) 11

13 where x R, k Σ E ; ˆvξ) = lim k,l) Θ E, k l = ξ Imk = Iml hk,l) for any ξ R, 5.7) where ˆv is defined by.1). Results of the type 5.6) go back to []. Results of the type 5.7) go back to [11]. These results follow, for example, from equation 5.4), formula 5.5) and the following estimates: gx,k) = O x 1 ) for x R, uniformly in k C \R, 5.8) Λ s gk)λ s L R ) L R ) = O k 1 ), for s > 1/, as k, k C \R, 5.9) where gx,k) is defined in 5.4), gk) denotes the integral operator with the Schwartz kernel gx y,k) and Λ denotes the multiplication operator by the function 1+ x ) 1/. Estimate 5.8) was given in [11]. Estimate 5.9) was formulated, first, in []. Concerning proof of 5.9), see [40]. In addition, estimate 5.9) in its zero energy restriction goes back to [9]. In the present work we use the following lemma: Lemma 5.1. Let satisfy 1.) and v satisfy 5.1). Let v L ) N for some N > 0. Then µx,k) c 5 1+N) for x R, k r 1+N), 5.10) where µx,k) is the Faddeev function of 5.4) and constants c 5,r > 0 depend on only. Lemma 5.1 is proved in Section 6. This proof is based on estimates 5.8) and 5.9). In addition, we have that see [7], [0]): h k,l) h 1 k,l) = π) R ψ 1 x, l)v x) v 1 x))ψ x,k)dx h k,l) h 1 k,l) = π) for k,l) Θ E, Imk = Iml = 0, and v 1, v satisfying 5.), ψ 1 x, l) [ˆΦ ˆΦ ) ] 1 ψ,k) x)dx for k,l) Θ E, Imk = Iml = 0, and v 1, v satisfying 1.5), 5.1), 5.11) 5.1) where ψ j, h j denote ψ and h of 5.) and 5.5) for v = v j, and ˆΦ j denotes the irichlet-to-neumann map ˆΦ for v = v j in, where j = 1,. In the present work we also use the following lemma: 1

14 Lemma 5.. Let satisfy 1.). Let v j satisfy 5.1), v j L ) N, j = 1,, for some N > 0. Then ˆv 1 ξ) ˆv ξ) h 1 k,l)+h k,l) c 6N1+N) v 1 v L ) E +ρ ) 1/ for k,l) Θ E, ξ = k l, Imk = Iml = ρ, E +ρ r 41+N), 5.1) where E R, ˆv j is the Fourier transform of v j, h j denotes h of 5.5) for v = v j, j = 1,) and constants c 6,r 4 > 0 depend on only. Some versions of estimate 5.1) were given in [16], [7], [0] see, for example, formula.18) of [16]). Lemma 5. is proved in Section 6. 6 Proofs of Lemmas.1, 5.1 and 5. Proof of Lemma 5.1. Using 5.1), 5.4) and 5.9), we get that µ,k) 1 L ) g y)vy)µy,k)dy R L ) c 7 N k µ,k) L ), 6.1) µ,k) L ) c +c 7 N k µ,k) L ), 6.) where c is defined by.15) and c 7 is some positive constant depending on only. Hence, we obtain that µ,k) L ) c for k c 7 N. 6.) We use also that 1 x y dy 1dy + x y 1 1 x y dy c 8, x, 6.4) where c 8 = c 8 ) > 0. Using 5.1), 5.4), 5.8), 6.), 6.4), we get that µx,k) 1+ gx y)vy)µy, k)dy 1/ 1+ gx y) dy N µ,k) L R ) c 5 )1+N) for x, k c 7 N. 6.5) 1

15 Proof of Lemma 5.. ue to 5.1), 5.11), we have that h k,l) h 1 k,l) = π) ψ 1 x, l)v x) v 1 x))ψ x,k)dx = = π) e ik l)x µ 1 x, l)v x) v 1 x))µ x,k)dx = 6.6) = ˆv k l) ˆv 1 k l)+i for k,l) Θ E, Imk = Iml = 0, where I = π) µ 1 x, l) 1)v x) v 1 x))µ x,k)dx+ +π) µ 1 x, l)v x) v 1 x))µ x,k) 1)dx+ +π) µ 1 x, l) 1)v x) v 1 x))µ x,k) 1)dx. Note that, for k,l) Θ E, E R, Imk = Iml = ρ, 6.7) k = Rek + Imk = k + Imk = E +ρ = l. 6.8) Using estimates 6.1), 6.), 6.5) in 6.7), we get that I π) µ 1, l) 1 L ) v v 1 L ) µ, l) L )+ + µ 1, l) L ) v v 1 L ) µ, l) 1 L )+ ) + µ 1, l) 1 L ) v v 1 L ) µ, l) 1 L ) c c 7 N v 1 v L )c 5 1+N) π) k + c 51+N) v 1 v L )c c 7 N π) + l + 1+c 51+N)) v 1 v L )c c 7 N π) l c 8 ) N1+N) v 1 v L ) E +ρ 6.9) for k,l) Θ E, Imk = Iml = ρ and k = l = E +ρ c 7 N. Formula 6.6) and estimate 6.9) imply 5.1). Proof of Lemma.1. ue to 5.11), we have that h k,l) h 1 k,l) c 9 ψ 1, l) L )δ ψ,k) L ), k,l) Θ E, Imk = Iml = 0, where c 9 = π) dx. 6.10) 14

16 Using formula 5.) and Lemma 5.1, we find that ψ j,k) L ) c 5 1+N)e Imk L, j = 1,, for k Σ E, k r 1+N), 6.11) where L = max x. Combining 6.8), 6.10) and 6.11), we get that x h k,l) h 1 k,l) c 9 c 51+N) e ρl δ, for k,l) Θ E, ρ = Imk = Iml, E +ρ r 1+N). 6.1) Note that foranyξ R satisfying ξ E +ρ whereρ > 0) thereexist some pair k,l) Θ E such that ξ = k l and Imk = Iml = ρ. Therefore, estimates 5.1) and 6.1) imply.). References [1] G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl.Anal. 7, 1988, [] N.V. Alexeenko, V.A. Burov, O.. Rumyantseva, Solution of threedimensional acoustical inverse scattering problem,ii: modified Novikov algorithm, Acoust. J. 54), 008, in Russian); English transl.: Acoust. Phys. 54), 008, [] R. Beals, R. Coifman, Multidimensional inverse scattering and nonlinear partial differential equations, Proc. Symp. Pure Math., 4, 1985, [4] L. Beilina, M.V. Klibanov, Approximate global convergence and adaptivity for coefficient inverse problems, Springer New York), pp. [5] A.P. Calderón, On an inverse boundary problem, Seminar on Numerical Analysis and its Applications to Continuum Physics, Soc. Brasiliera de Matematica, Rio de Janeiro, 1980, [6] M. i Cristo, L. Rondi Examples of exponential instability for inverse inclusion and scattering problems Inverse Problems 19, 00, [7] L.. Faddeev, Growing solutions of the Schrödinger equation, okl. Akad. Nauk SSSR, 165, N., 1965, in Russian); English Transl.: Sov. Phys. okl. 10, 1966, [8] L.. Faddeev, The inverse problem in the quantum theory of scattering. II, Current problems in mathematics, Vol., 1974, pp , 59. Akad. Nauk SSSR Vsesojuz. Inst. Naucn. i Tehn. Informacii, Moscowin Russian); English Transl.: J.Sov. Math. 5, 1976, [9] I.M. Gelfand, Some problems of functional analysis and algebra, Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, pp

17 [10] P. Hähner, T. Hohage, New stability estimates for the inverse acoustic inhomogeneous medium problem and applications, SIAM J. Math. An., ), 001, [11] G.M. Henkin and R.G. Novikov, The -equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk 4), 1987, 9 15 in Russian); English Transl.: Russ. Math. Surv. 4), 1987, [1] M.I. Isaev, Exponential instability in the Gel fand inverse problem on the energy intervals, J. Inverse Ill-Posed Probl., Vol. 19), 011, [1] M.I. Isaev, Instability in the Gel fand inverse problem at high energies, Applicable Analysis, Vol. 9, No. 11, 01, [14] M.I. Isaev, Energy and regularity dependent stability estimates for near-field inverse scattering in multidimensions, Journal of Mathematics, Article I 18154, 01, 10 p.; OI: /01/ [15] M.I. Isaev, Exponential instability in the inverse scattering problem on the energy interval, Func. Anal. Prilozh., Vol. 47), 01, 8 6 in Russian); Engl. translation: Functional Analysis and Its Applications, Vol. 47), 01, [16] M.I. Isaev, R.G. Novikov Energy and regularity dependent stability estimates for the Gel fand inverse problem in multidimensions, J. of Inverse and Ill-posed Probl., Vol. 0), 01, 1 5. [17] M.I. Isaev, R.G. Novikov, Stability estimates for determination of potential from the impedance boundary map, Algebra and Analysis, Vol. 51), 01, 7 6 in Russian); Engl. Transl.: St. Petersburg Mathematical Journal, Vol. 5, 014, 41. [18] M.I. Isaev, R.G. Novikov, Reconstruction of a potential from the impedance boundary map, Eurasian Journal of Mathematical and Computer Applications, Vol. 11), 01, 5 8. [19] M.I. Isaev, R.G. Novikov, New global stability estimates for monochromatic inverse acoustic scattering, SIAM Journal on Mathematical Analysis, Vol. 45), 01, [0] V. Isakov, Increasing stability for the Schrödinger potential from the irichlet-to-neumann map, iscrete Contin. yn. Syst. Ser. S 4, 011, no., [1] V. Isakov, S. Nagayasu, G. Uhlmann, J.-N. Wang, Increasing stability of the inverse boundary value problem for the Schrödinger equation, Contemporary Mathematics to appear), e-print arxiv: [] R.B. Lavine and A.I. Nachman, On the inverse scattering transform of the n-dimensional Schrödinger operator Topics in Soliton Theory and Exactly Solvable Nonlinear Equations ed M Ablovitz, B Fuchssteiner and M Kruskal Singapore: World Scientific), 1987, pp

18 [] M.M. Lavrentev, V.G. Romanov, S.P. Shishatskii, Ill-posed problems of mathematical physics and analysis, Translated from the Russian by J. R. Schulenberger. Translation edited by Lev J. Leifman. Translations of Mathematical Monographs, 64. American Mathematical Society, Providence, RI, vi+90 pp. [4] N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems 17, 001, [5] S. Nagayasu, G. Uhlmann, J.-N. Wang, Increasing stability in an inverse problem for the acoustic equation, Inverse Problems 9, 01, pp). [6] R.G. Novikov, Multidimensional inverse spectral problem for the equation ψ + vx) Eux))ψ = 0, Funkt. Anal. Prilozhen. 4), 1988, 11- in Russian); Engl. Transl. Funct. Anal. Appl., 1988, 6-7. [7] R.G. Novikov, -method with nonzero background potential. Application to inverse scattering for the two-dimensional acoustic equation, Comm. Partial ifferential Equations 1, 1996, no. -4, [8] R.G. Novikov, Rapidly converging approximation in inverse quantum scattering in dimension, Physics Letters A 8, 1998, [9] R.G. Novikov, The -approach to approximate inverse scattering at fixed energy in three dimensions. IMRP Int. Math. Res. Pap. 005, no. 6, [0] R.G. Novikov, Formulae and equations for finding scattering data from the irichlet-to-neumann map with nonzero background potential, Inverse Problems 1, 005, [1] R.G. Novikov, The -approach to monochromatic inverse scattering in three dimensions, J. Geom. Anal 18, 008, [] R.G. Novikov, An effectivization of the global reconstruction in the Gel fand-calderon inverse problem in three dimensions, Contemporary Mathematics, 494, 009, [] R.G. Novikov, New global stability estimates for the Gel fand-calderon inverse problem, Inverse Problems 7, 011, pp). [4] R.G. Novikov, Approximate Lipschitz stability for non-overdetermined inverse scattering at fixed energy, J.Inverse Ill-Posed Probl., Vol. 16), 01, [5] R. Novikov and M. Santacesaria, A global stability estimate for the Gelfand- Calderon inverse problem in two dimensions, J.Inverse Ill-Posed Probl., Volume 18, Issue 7, 010, [6] L. Päivärinta, M. Zubeldia, The inverse Robin boundary value problem in a half-space, e-print arxiv: [7] M. Santacesaria, Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions, Applicable Analysis, 01, Vol. 9, No. 8,

19 [8] M. Santacesaria, A Holder-logarithmic stability estimate for an inverse problem in two dimensions, J. Inverse Ill-Posed Probl. to appear), e-print arxiv: [9] J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. 15, 1987, [40] R. Weder, Generalized limiting absorption method and multidimensional inverse scattering theory, Mathematical Methods in the Applied Sciences, 14, 1991, M.I. Isaev Centre de Mathématiques Appliquées, Ecole Polytechnique, 9118 Palaiseau, France Moscow Institute of Physics and Technology, olgoprudny, Russia isaev.m.i@gmail.com R.G. Novikov Centre de Mathématiques Appliquées, Ecole Polytechnique, 9118 Palaiseau, France Institute of Earthquake Prediction Theory and Math. Geophysics RAS, Moscow, Russia novikov@cmap.polytechnique.fr 18