Uniqueness for the inverse fixed angle scattering problem

Juan Antonio Barceló; Carlos Castro; Teresa Luque; Cristobal J. Meroño; Alberto Ruiz; María de la Cruz Vilela

doi:10.1515/jiip-2019-0019

Published by De Gruyter March 10, 2020

Uniqueness for the inverse fixed angle scattering problem

Juan Antonio Barceló , Carlos Castro , Teresa Luque , Cristobal J. Meroño , Alberto Ruiz and María de la Cruz Vilela

From the journal Journal of Inverse and Ill-posed Problems

https://doi.org/10.1515/jiip-2019-0019

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Abstract

We present a uniqueness result in dimensions 3 for the inverse fixed angle scattering problem associated to the Schrödinger operator -Δ+q, where q is a small real-valued potential with compact support in the Sobolev space Wβ,2, with β>0. This result improves the known result [P. Stefanov, Generic uniqueness for two inverse problems in potential scattering, Comm. Partial Differential Equations 17 1992, 55–68], in the sense that almost no regularity is required for the potential. The uniqueness result still holds in dimension 4, but for more regular potentials in Wβ,2, with β>2/3. The proof is a consequence of the reconstruction method presented in our previous work, [J. A. Barceló, C. Castro, T. Luque and M. C. Vilela, A new convergent algorithm to approximate potentials from fixed angle scattering data, SIAM J. Appl. Math. 78 2018, 2714–2736].

Keywords: Inverse scattering problem; Schrödinger equation; scattering data

MSC 2010: 35P25; 35R30; 35J05

Funding statement: The first, the second, the fourth and the sixth author were supported by Spanish Grant MTM2017-85934-C3-3-P, the third by Spanish Grant MTM2017-82160-C2-1-P, and the fifth by Spanish Grant MTM2017-85934-C3-2-P.

A Appendix

In this section we detail a gap in the proof of the global uniqueness for the fixed angle scattering problem given in [9]. Essentially, the main result in [9, Theorem 1.1] follows from putting together key estimates (25) and (26) in that paper.

Estimate (26) (which coincides with estimate (14) below) is proved in [9, Lemma 3.1] that we now state.

Lemma 7.

Assume that f is a real function compactly supported in B⁢(0,R) and such that f∈Wβ,2⁢(B⁢(0,R)), with β>3. Let κ>0 be fixed and η>0. Then

lim supη→∞maxζ∈𝕊2|f^((κ+iη)ζ)|=∞.

Also, for any κ>0, there is an η=η⁢(κ) such that

(14)maxζ∈𝕊2|f^((κ+iη)ζ)|≥maxξ∈ℝ3|f^(ξ)|

and

η⁢(κ)=R-1⁢ln⁡κ+O⁢(1),κ→∞.

Our main objection is that (14) cannot be true for a C0∞ function and a sequence of the kind η⁢(κ)=O⁢(ln⁡κ). In fact, if f∈C0∞⁢(B⁢(0,R)), by Paley–Wiener (see [4, p. 21]), we have that

|f^⁢((κ+i⁢η)⁢ζ)|≤C⁢(γ)⁢|κ+i⁢η|-γ⁢eR⁢|η|

holds for every γ>0. Then it is clear that the condition η⁢(κ)=O⁢(ln⁡κ) as κ→∞ implies that the left hand side in (14) tends to 0 as κ→∞, contradicting the inequality.

Since the logarithmic behavior of η⁢(κ) is a condition to prove estimate (25) (see [9, Estimates (45)–(49)]), both estimates (25) and (26) cannot be used simultaneously together to prove [9, Theorem 1.1].

References

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Received: 2019-02-27

Revised: 2019-07-31

Accepted: 2020-01-31

Published Online: 2020-03-10

Published in Print: 2020-08-01

Uniqueness for the inverse fixed angle scattering problem

Abstract

A Appendix

Lemma 7.

References

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