We wish to determine properties of either potential V or obstacle B from knowledge of only the scattering matrixSV(k) orSB(k). The good thing is that this can be done! This has not been the main focus of the research undertaken for this thesis, so we shall just state results and give a sketch of how these results are proven, along with commentary on the physical validity of the theorems.
3.3.1 Potential Scattering Inverse Problem
Theorem 3.9 ([Mel95], Corollary 3.1). The map V 7→ SV from (complex-valued) potentials V ∈
Cc∞(Rn) to scattering matrix, is injective.
The proof of this theorem requires analysing k → ∞ asymptotics of the smooth kernel of
SV(k)−I. It can be shown that this integral kernel has an asymptotic expansion in k ask→ ∞, and the coefficients of the expansion are determined from V via its X-ray transform, which essentially measures averages of V along rays from the origin. The X-ray transform is invertible, and so V can be recovered from the asymptotic expansion.
3.3. INVERSE SCATTERING PROBLEM 31
Some benefits of the proof of this theorem include that it gives a way of actually reconstructing the potential from the scattering matrix. But the proof requires high frequency knowledge of the scattering matrix, which corresponds in our physical problems to either causing tsunamis on the shore of a lake or blasting somebodies head with high energy radiation, both of which aren’t physically feasible. A proof of Theorem3.9can be found in [Mel95], in which it is labelled Corollary 3.1.
As an alternative, we have the following:
Theorem 3.10 ([Mel95], Proposition 3.5). For any n ≥ 3 and any k > 0, the map from real potentials V ∈Cc∞ to scattering matrix at frequency k, SV(k), is injective.
Before explaining the idea behind the proof of this theorem, we shall examine its pros and cons in comparison to the previous theorem. The dimensional requirement is fine in our physical problems, since the universe has spatial dimension at least 3. The reduction to real potentials is also fine for the case of most measurements in our universe. That scattering at only one frequency determines the potential is a clear benefit over the last theorem in which knowledge for asymptotic frequencies was necessary. However, this theorem has the (major) downside that it is not re-constructive: it does not produce a method of determining the potential from the scattering matrix. The proof of this theorem starts by assuming that two real valued potentials V1, V2 ∈ Cc∞ have the same scattering matrix. It can then be shown that for any g∈C∞(Sn−1), the following holds:
0 = Z
Rn
(V1(x)−V2(x))(HV1g)(x)(HV2g)(x)
where HV1,2 are as in Equation 3.7. It is then shown that in dimension n≥ 3, for any ξ ∈ R
n, a
g can be chosen so that (HV1g)(x)(HV2g)(x) approximates e
ix·ξ to arbitrary precision. Then the above equality with these chosen g implies that in the limit the Fourier transforms of V1 and V2
agree. Since both V1 and V2 are compactly supported they are both in L2(Rn), in which case we may invert the Fourier transform to find V1 = V2 almost everywhere. Smoothness of V1 and V2
implies equality everywhere. This provides no technique for reconstruction of V. For a complete proof see [Mel95], in which this theorem is labelled Proposition 3.5.
3.3.2 Obstacle Scattering Inverse Problems
As in the potential scattering case, we first have a theorem about injectivity of the mapping from obstacle to scattering matrix
Theorem 3.11 ([Mel95], Proposition 5.2). Supposen≥3is odd andB1, B2 are two obstacles with
smooth boundary. Then if there exists f ∈ C∞(Sn−1) not identically zero such that SB1(k)f =
SB2(k)f for all k >0, then B1=B2.
As in the second inverse scattering theorem given for potentials, this theorem does not give a method of reconstructing the obstacle from the scattering. It also requires more stringent con- straints on the dimension of the space we work over. The parity constraint is incurred during the proof of existence of the perturbed plane wave solutions in Lemma 3.3, in which the resolvent of
−∆
Rn\B−k
2 must be continuously extended to real k. This was a detail skipped earlier in this
thesis as it is quite complicated and not very informative. For more information, see the Chapter 2 of [Mel95]. However, this condition is satisfied by the (classical) universe we live in, so does not produce an issue with most applications of this mathematical framework to the real world.
As an advantage over the potential scattering inverse problem theorems, this theorem only requires checking one incoming wave, the f ∈ C∞(Sn−1) mentioned in the requirements. This is nice for real world applications, as your favourite scanning machine need only produce one waveform. However, the condition that SB1(k)f =SB2(k)f must hold for allk >0 is a significant
issue as in the potential case: physically this corresponds again to either causing tidal waves at the side of a lake or bombarding a person’s head with high-energy radiation.
There has been significant work put into reconstructive inverse obstacle scattering problems, due to their applicability to the real world. It was shown by David Colton in [Col84] that for general data there may be no obstacle producing this data as scattering data, and even if the data does correspond to an obstacle, this correspondence is in no way continuous. This lack of continuity makes the typical methods of analysis impossible to use, as approximation requires continuity. For this reason, it seems that most work towards inverse obstacle scattering has been focused on numeric approximation, given some prior assumed knowledge of the obstacle.
For example, in [Has11] a method is used in which it is assumed that an obstacleDlies within some chosen and nice bounded open setG, with conditions onGconcerning the spectrum of−∆G. The waves scattered off D with Dirichlet boundary conditions can be approximated outside G. These approximations are then approximated further in G\D, and then the boundary of D is found as the zero level set of this further approximation. The methods of approximation have some serious issues with non-convexity in regions of the boundary of the obstacle D. There is another method considered in [Has11] in which a certain modified form of the Poisson kernel which blows up on the boundary of the obstacle is investigated, which can be partially reconstructed from the scattering data. Both of these methods seem to give good approximations of the obstacle shape at least at a qualitative level, based on the diagrams shown in the same paper.