Exponential instability in the inverse scattering problem on the energy intervals

Transcription

1 ECOLE POLYTECHNIQUE CENTRE DE MATHÉMATIQUES APPLIQUÉES UMR CNRS PALAISEAU CEDEX FRANCE). Tél: Fax: Exponential instability in the inverse scattering problem on the energy intervals Mikhail Isaev R.I. 70 December 00

2

3 Exponential instability in the inverse scattering problem on the energy intervals M.I. Isaev Moscow Institute of Physics and Technology, 4700 Dolgoprudny, Russia Centre de Mathématiqu Appliqué, Ecole Polytechnique, 98 Palaiseau, France Abstract We consider the inverse scattering problem on the energy interval in three dimensions. We are focused on stability and instability qutions for this problem. In particular, we prove an exponential instability timate which shows optimality of the logarithmic stability rult of [Stefanov, 990] up to the value of the exponent). Introdution We consider the Schrödinger equation where ψ + vx)ψ = Eψ, x R 3,.) v is real-valued, v L R 3 ), vx) = O x 3 ɛ ), x, for some ɛ > 0..) Under conditions.), for any k R 3 \ 0 equation.) with E = k has a unique continuous solution ψ + x, k) with asymptotics of the form ) ) ψ + x, k) = e ikx π ei k x k f x k, x x, k + o x as x uniformly in x ), x.3) where fk/ k, ω, k ) with fixed k is a continuous function of ω S. The function fθ, ω, s) arising in.3) is refered to as the scattering amplitude for the potential v for equation.). For more information on direct scattering for equation.), under condition.), see, for example, [6] and [].) It is well known that for equation.), under conditions.), the scattering amplitude f in its high-energy limit uniquely determin ˆv on R 3, where ˆvp) = π) 3 R 3 e ipx vx)dx, p R 3,.4) via the Born formula. As a mathematical theorem this rult go back to [5] see, for example, Section. of [] and Theorem. of [4] for details). We consider the following inverse problem for equation.).

4 Problem. Given f on the energy interval I, find v. In [7] it was shown that for equation.), under the conditions.), for any E > 0 and δ > 0 the scattering amplitude fθ, ω, s) on {θ, ω, s) S S R +, E s E + δ} uniquely determin ˆvp) on {p R 3 p E}. This determination is based on solving linear integral equations and on an analytic continuation. This rult of [7] was improved in [4]. On the other hand, if v satisfi.) and, in addition, is compactly supported or exponentially decaying at infinity, then ˆvp) on {p R 3 p E} uniquely determin ˆvp) on {p R 3 p > E} by an analytic continuation and, therefore, uniquely determin v on R 3. In the case of fixed energy and potential v, satisfying.) and, in addition, being compactly supported or exponentially decaying at infinity, global uniquens theorems and precise reconstructions were given for the first time in [], [3]. An approximate but numerically efficient method for finding potential v from the scattering amplitude f in the case of fixed energy was devoloped in [5]. Related numerical implementation was given in []. Global stability timat for Problem. were given by Stefanov in [7] at fixed energy for compactly supported potentials), see Theorem. in Section of the prent paper. In [7], using a special norm for the scattering amplitude f, it was shown that the stability timat for Problem. follow from the Alsandrini stability timat of [] for the Gel fand-calderon inverse problem of finding potential v in bounded domain from the Direchlet-to-Neumann map. The Alsandrini stability timat were recently improved by Novikov in [6]. In the case of fixed energy, the Mandache rults of [0] show that logarithmic stability timat of Alsandrini of [] and pecially of Novikov of [6] are optimal up to the value of the exponent). In [8] studi of Mandache were extended to the case of Direchlet-to-Neumann map given on the energy intervals. Note also that Mandache-type instability timat for the elliptic inverse problem concerning the determination of inclusions in a conductor by different kinds of boundary measurements and the inverse obstacle acoustic scattering problems were given in [3]. In the prent work we apply to Problem. the approach of [0], [8] and show that the Stefanov logarithmic stability timat of [7] are optimal up to the value of the exponent). The Stefanov stability timat and our instability rult for Problem. are prented and discussed in Section. In Section 3 we prove some basic analytic properti of the scattering amplitude. Finally, in Section 5 we prove the main rult, using a ball packing and covering by ball arguments. Stability and instability timat In what follows we suppose supp vx) D = B0, ),.)

5 where Bx, r) is the open ball of radius r centered at x. We consider the orthonormal basis of the spherical harmonics in L S ) = L D): {Y p j : j 0; p j + }..) The notation a jp j p ) stands for a multiple sequence. We will drop the subscript 0 j, p j +, 0 j, p j +..3) We expand function fθ, ω, s) in the basis {Y p j fθ, ω, s) = As in [7] we use the norm f,, s) σ,σ = j,p,j,p a jp j p s)y p j j,p,j,p j + Y p j }: ) j+σ j + θ)y p j ω)..4) ) j+σ a jpjp s).5) If a function f is the scattering amplitude for some potential v L D) supported in B0, ρ), where 0 < ρ <, then /. a jp j p s) Cs, v L D)) ρ ) j+3/ ρ ) j+3/ j + j +.6) and, therefore, f,, s) σ,σ <, see timat of Proposition. of [7]. Theorem. see [7]). Let v, v be real-valued potentials such that v i L D) H q R 3 ), supp v i B0, ρ), v i L D) N for i =, and some N > 0, q > 3/ and 0 < ρ <. Let f and f denote the scattering amplitud for v and v, rpectively, in the framework of equation.) with E = s, s > 0, then v v L D) cn, ρ)φ δ f,, s) f,, s) 3/, / ),.7) where φ δ t) = ln t) δ for some fixed δ, where, in particular, 0 < δ <, and for sufficiently small t > 0. The main rult of the prent work is the following theorem. Theorem.. For the interval I = [s, s ], such that s > 0, and for any m > 0, α > m and any real σ, σ there are constants β > 0 and N > 0, such that for any v 0 C m D) with v 0 L D) N, supp v 0 B0, /) and any ɛ 0, N), there are real-valued potentials v, v C m D), also supported in B0, /), such that sup f,, s) f,, s) σ,σ ) exp s I v v L D) ɛ, v i v 0 L D) ɛ, i =,, v i v 0 C m D) β, i =,, 3 ) ɛ α,.8)

6 where f, f are the scattering amplitud for v, v, rpectively, for equation.). Remark.. In the case of fixed energy s = s we can replace the condition α > m in Theorem. by α > 5m/3. Remark.. We can allow β to be arbitrarily small in Theorem. if we require ɛ ɛ 0 and replace the right-hand side in the first inequality in.8) by exp cɛ α ), with ɛ 0 > 0 and c > 0 depending on β. Remark.3. Note that Theorem. and Remark. imply, in particular, that for any real σ and σ the timate v v L D) cn, ρ, m, I) sup φ δ f,, s) f,, s) σ,σ ).9) s I can not hold with δ > m in the case of the scattering amplitude given on the energy interval and with δ > 5m/3 in the case of fixed energy. Thus Theorem. and Remark. show optimality of the Stefanov logarithmic stability rult up to the value of the exponent). Remark.4. A disadvantage of timate.7) is that δ < even if m is very great..0) Apparently, proceeding from rults of [6], it is not difficult to improve timate.7) for δ = m + om) as m..) 3 Some basic analytic properti of the scattering amplitude Consider the solution ψ + x, k) of equation., see formula.3). We have that ψ + x, k) = e ikx µ + x, θ, s), 3.) where θ S, k = sθ and µ + x, θ, s) solv the equation µ + x, θ, s) = G + x, y, s)e isθx y) vy)µ + y, θ, s)dy, 3.) R 3 where G + x, y, s) = eis x y 4π x y. 3.3) We suppose that condition.) holds and, in addition, for some h > 0 we have that Im s h, 3.4) c h, D) v L D) /, 3.5) 4

7 where D = B0, ), e h x y c h, D) = sup dy. 3.6) x D D 4π x y Then, in particular, e isθx y) e is x y e h x y. 3.7) Solving 3.) by the method of succive approximations in L D), we obtain that µ + x, θ, s), θ S, x D. 3.8) c v L D) Lemma 3.. Let a jp j p s) denote coefficients fs, θ, ω) in the basis of the spherical harmonics {Y p j Y p j }, where f is the scattering amplitude for potential v L D) such that conditions.) and 3.5) hold for some h > 0, fθ, ω, s) = a jp j p s)y p j θ)y p j ω). 3.9) j,p,j,p Then a jp j p s) is holomorphic function in W h = {s Im s h} and a jp j p s) c h, D) for s W h. 3.0) Proof of Lemma 3.. We start with the well-known formula fθ, ω, s) = π) 3 e isθ ω)x vx)µ + x, θ, s)dx. 3.) R 3 Note that, since θ, ω S, e isθ ω)x e Im s x. 3.) Combining it with.), 3.5), 3.8) and 3.) we obtain that Using also that a jp j p s) = we obtain the rult of Lemma 3.. fθ, ω, s) c h, D) for s W h. 3.3) fθ, ω, s)y p j S S θ)y p j ω)dθdω 3.4) 4 A fat metric space and a thin metric space Definition 4.. Let X, dist) be a metric space and ɛ > 0. We say that a set Y X is an ɛ-net for X X if for any x X there is y Y such that distx, y) ɛ. We call ɛ-entropy of the set X the number H ɛ X ) := log min{ Y : Y is an ɛ-net fot X }. A set Z X is called ɛ-discrete if for any distinct z, z Z, we have distz, z ) ɛ. We call ɛ-capacity of the set X the number C ɛ := log max{ Z : Z X and Z is ɛ-discrete}. 5

8 The use of ɛ-entropy and ɛ-capacity to derive properti of mappings between metric spac go back to Vitushkin and Kolmogorov see [9] and referenc therein). One notable application was HilbertŠs 3th problem about reprenting a function of several variabl as a composition of functions of a smaller number of variabl). In sence, Lemma 4. and Lemma 4. are parts of the Theorem XIV and the Theorem XVII in [9]. Lemma 4.. Let d è m > 0. For ɛ, β > 0, consider the real metric space X mɛβ = {v C m R d ) supp v B0, /), v L R d ) ɛ, v C m R d ) β} with the metric induced by L. Then there is µ > 0 such that for any β > 0 and ) ɛ 0, µβ), there is an ɛ-discrete set Z X mɛβ with at least exp d µβ/ɛ) d/m elements. Lemma 4.. For the interval I = [a, b] and γ > 0 consider the ellipse W I,γ C: W I,γ = { a + b + a b cos z Im z γ}. 4.) Then there is a constant ν = νc, γ) > 0 such that for any δ 0, e ) there is a δ-net for the space of functions on I with L -norm, having holomorphic continuation to W I,γ with module bounded above on W I,γ by the constant C, with at most expνln δ ) ) elements. Remark 4.. In the case of a = b, taking Y = δ Z [ C, C] + i δ Z [ C, C], 4.) we get δ-net with at most expν ln δ ) elements. Lemma 4. and Lemma 4. were also formulated and proved in [0] and [8], rpectively. For the interval I = [s, s ] such that s > 0 and real σ, σ we introduce the Banach space { ) } X I,σ,σ = a s) jpjp a jp j p s) <, 4.3) where a jp j p s) ) XI,σ,σ = sup j,p,j,p s I ) XI,σ,σ j ) j+σ + ) ) j+σ j + a jpjp s). 4.4) We consider the scattering amplitude f for some potential v L D) supported in B0, ρ), where 0 < ρ <. We identify ) in the sequel the scattering amplitude fs, θ, ω) with its matrix a jp j p s) in the basis of the spherical harmonics {Y p j Y p j }. We have that sup f,, s) σ,σ c 3 a jp j p s) s I 6 ) XI, σ, σ, 4.5)

9 where σ σ = σ σ = 3 and c 3 = c 3 I) >. We obtain 4.5) from definitions.5), 4.4) and by taking c 3 > in a such a way that j,p,j,p ) 3 ) 3 j + j + < c ) For h > 0 we denote by A h the set of the matric, corrponding to the scattering amplitud for the potentials v L D) supported in B0, /) such that condition 3.5) holds. Lemma 4.3. For any h > 0 and any real σ, σ, the set A h belongs to X I,σ,σ. In addition, there is a constant η = ηi, h, σ, σ ) > 0 such that for any δ 0, e ) there is a δ-net Y for A h in X I,σ,σ with at most exp η ln δ ) 6 ) ) + ln ln δ elements. Proof of Lemma 4.3. We can suppose that σ, σ 0 as the assertion is stronger in this case. If a function f is the scattering amplitude for some potential v L D) supported in B0, /), we have from.6) that ) j+σ j + ) j+σ j + j + ) σ j + ) σ a jpjps) c 4, j+j 4.7) where c 4 = c 4 I, h) > 0. Hence, for any positive σ and σ, a jp j p s) ) XI,σ,σ sup j,j c 4 j + ) σ j + ) σ j+j ) < 4.8) and so the first assertion of the Lemma 4.3 is proved. Let l δ,σ,σ be the smallt natural number such that c 4 l + ) σ+σ l < δ for any l l δ,σ,σ. Taking natural logarithm we have that ln c 4 σ + σ ) lnl + ) + l ln > ln δ for l l δ,σ,σ. 4.9) Using ln δ >, we get that l δ,σ,σ C ln δ, 4.0) where the constant C depends only on h, σ, σ and I = [s, s ]. We take W I = W I,γ of 4.), where the constant γ > 0 is such that W I {s Im s h}. If maxj, j ) l δ,σ,σ, then we denote by Y jp j p some δ jp j p -net from Lemma 4. with the constant C = c, where the constant c is from Lemma 3. and ) j+σ ) j+σ δ jp j p = δ. 4.) j + j + Otherwise we take Y jp j p = {0}. We set { ) } Y = a jp j p s) a jp j p s) Y jp j p. 4.) 7

10 ) ) For any a jp j p s) A h there is an element b jp j p s) Y such that ) j+σ j + ) j+σ j + a jpjp s) b s) jpjp ) j+σ j + ) j+σ j + 4.3) δ jpjp δ in the case of maxj, j ) l δ,σ,σ and ) j+σ j + ) j+σ j + a jpjps) b jpjps) j + ) σ j + ) σ maxj, j ) + ) σ+σ c 4 c 4 < δ, j+j maxj,j) 4.4) otherwise. It remains to count the elements of Y. We recall that Y jp j p = in the case of maxj, j ) > l δ,σ,σ. Using again the fact that ln δ and 4.0) we get in the case of maxj, j ) l δ,σ,σ : Y jp j p expνln δ j p j p ) ) exp ν ln δ ) + ln ln δ ) ). 4.5) We have that n δ,σ,σ lδ,σ,σ l δ,σ,σ + ) l δ,σ,σ + ) 4, where n δ,σ,σ is the number of four-tupl j, p, j, p ) with maxj, j ) l δ,σ,σ. Taking η to be big enough we get that Y exp ν ln δ ) + ln ln δ ) )) n δ,σ,σ ν ln δ ) + ln ln δ ) + C ln δ ) 4) exp exp η ln δ ) 6 + ln ln δ ) ). 4.6) Remark 4.. In the case of s = s, taking into account Remark 4. and using it in 4.5) and 4.6), we get δ-net Y with at most exp η ln δ ) 5 )) + ln ln δ elements. 5 Proof of Theorem. We take N such that condition 3.5) holds for any v L D) N for some h > 0. By Lemma 4., the set v 0 + X mɛβ has an ɛ-discrete subset v 0 + Z. Since ɛ 0, N) we have that the set Y constructed in Lemma 4.3 is also δ-net for the set of the matric, corrponding to the scattering amplitud for the potentials 8

11 ) v v 0 + X mɛβ. We take δ such that c 3 δ = exp ɛ α, see 4.5). Note that inequaliti of.8) follow from v 0 + Z > Y, 5.) where the set Y is constructed in Lemma 4.3 with σ = σ + 3 and σ = σ + 3. In fact, if v 0 + Z > Y, ) then there are two ) potentials v, v v 0 + Z with the matric a jp j p s) and b jp j p s), corrponding to the scattering amplitud for them, being in the same X I,σ,σ -ball radius δ centered at a point of Y. Hence, using 4.5) we get that ) sup f,, s) f,, s) σ,σ c 3 a jp j p s) s I ) c 3 δ = exp ɛ α. ) b jp j p s) XI, σ, σ 5.) It remains to find β such that 5.) is fullfiled. By Lemma 4.3 for some η α = η α I, σ, σ, α) > 0 ) 6 )) ) ) Y exp η lnc 3 ) + ɛ α + ln lnc 3 ) + ɛ α exp η α ɛ 3 m. Now we take 5.3) β > µ max N, ηα m/3 m). 5.4) This fulfils requirement ɛ < µβ in Lemma 4., which giv v 0 + Z = Z exp 4 µβ/ɛ) 3/m) 5.4) > > exp 4 ηα m/3 m /ɛ) 3/m) 5.3) Y. 5.5) This complet the proof of Theorem.. In the case of fixed energy s = s, using Remark 4. in 5.3), we can replace the condition α > m in Theorem. by α > 5m/3. Acknowledgments This work was fulfilled under the direction of R.G.Novikov in the framework of an internship at Ecole Polytechnique. Referenc [] G.Alsandrini, Stable determination of conductivity by boundary measurements, Appl.Anal )

12 [] N.V. Alexeenko, V.A. Burov and O.D. Rumyantseva, Solution of threedimensional acoustical inverse scattering problem,ii: modified Novikov algorithm, Acoust. J. 543) 008) in Russian), English transl.: Acoust. Phys. 543) 008). [3] M. Di Cristo and L. Rondi Exampl of exponential instability for inverse inclusion and scattering problems Inverse Problems ) [4] I.M. Gelfand, Some problems of functional analysis and algebra, Proceedings of the International Congrs of Mathematicians, Amsterdam, 954, pp [5] L.D. Faddeev,Uniquens of the solution of the inverse scattering problem. Vtn. Leningr. Univ in Russian), 956. [6] L.D. Faddeev, The inverse problem in the quantum theory of scattering. II. Russian) Current problems in mathematics, Vol. 3 Russian), pp , 59. Akad. Nauk SSSR Vsojuz. Inst. Naucn. i Tehn. Informacii, Moscow, 974. [7] G.M. Henkin and R.G. Novikov, The -equation in the multidimensional inverse scattering problem, Uspekhi Mat. Nauk 43) 987), 93-5 in Russian); English Transl.: Russ. Math. Surv. 43) 987), [8] M.I. Isaev, Exponential instability in the Gel fand inverse problem on the energy intervals, e-print arxiv: [9] A.N. Kolmogorov, V.M. Tikhomirov, ɛ-entropy and ɛ-capacity in functional spac Usp. Mat. Nauk 4959) 3-86 in Russian) Engl. Transl. Am. Math. Soc. Transl. 7 96) ) [0] N. Mandache, Exponential instability in an inverse problem for the Schrödinger equation Inverse Problems. 700) [] R.G. Newton, Inverse Schrodinger scattering in three dimensions. Texts and Monographs in Physics. Springer-Verlag, Berlin, 989. x+70 pp. [] R.G. Novikov, Multidimensional inverse spectral problem for the equation ψ + vx) Eux))ψ = 0 Funkt. Anal. Prilozhen. 4)988) - in Russian) Engl. Transl. Funct. Anal. Appl. 988) 63-7). [3] R.G. Novikov, The inverse scattering problem at fixed energy for the threedimensional Schrodinger equation with an exponentially decreasing potential. Comm. Math. Phys ), no. 3, [4] R.G. Novikov, On determination of the Fourier transform of a potential from the scattering amplitude. Inverse Problems 7 00), no. 5,

13 [5] R.G. Novikov, The -approach to approximate inverse scattering at fixed energy in three dimensions. IMRP Int. Math. R. Pap. 005, no. 6, [6] R.G. Novikov, New global stability timat for the Gel fand-calderon inverse problem, Inverse Problems 7 pp) 0500; e-print arxiv: [7] P. Stefanov, Stability of the inverse problem in potential scattering at fixed energy Annal de l institut Fourier, tome 40, N4 990), p