Remarks on the Phaseless Inverse Uniqueness of a Three-Dimensional Schrödinger Scattering Problem

We consider the inverse scattering theory of the Schrödinger equation. The inverse problem is to identify the potential scatterer by the scattered waves measured in the far-fields. In somemicro/nanostructures, it is impractical to measure the phase information of the scattered wave field emitted from the source. We study the asymptotic behavior of the scattering amplitudes/intensity from the linearization theory of the scattered wave fields.The inverse uniqueness of the scattered waves is reduced to the inverse uniqueness of the analytic function. We deduce the uniqueness of the Schrödinger potential via the identity theorems in complex analysis.

The phaseless inverse scattering problem (PISP) [1][2][3][4][5][6] is to find the Schrödinger potential () if the information is given or partially given on the following types of scattering data measured in far-fields: (PISP 1) :  (, , ) fl       (, , )     2 , (, , ) ∈  × C × ; (5) We call the inverse problem given (5) the phaseless inverse scattering problem of type 1 and (6) type 2. This is a longstanding open problem in the inverse scattering theory of quantum mechanics [7][8][9][10][11], and the breakthrough is made in 2 Advances in Mathematical Physics a series of papers by 12].The problem is typical in quantum mechanics and in the study of nano/microstructures when it is impractical to measure the phase of the scattered waves or the signals.The problem is also common in the inverse problems in micro/nanostructures, for example, electron microscopy, crystallography, medical imaging, and nanooptics.In crystallography, the phase information of a scattered wave field may be recovered if the intensity pattern at and between the Bragg peaks of diffracted wave is finely measured [13].Additionally, due to the quality of many optical equipment and lens-like devices, the measurements may suffer from very serious optical aberration when the index of refraction is close to one.The engineering economy is that the phase retrieval algorithms may not be limited by the quality of the optical equipment.
In astronomy, the research objects are usually distant stars, which are optically incoherent sources.In the cases of incoherent waves, the phase is stochastic.Hence, the optical signal is received/conceived as the intensity of the light, the square modulus of the complex-valued wave field, or the square modulus of the related Fourier transform.Moreover, the measurements usually are inflicted with very serious optical aberration caused either by atmospheric turbulence or by the imperfection of the optical imaging system.Thus, we ask if it is sufficient to reconstruct the scatterer/image source using only the intensity of the scattered wave fields or the signals.
Let us state two inverse uniqueness results in this paper.Here we provide similar results to the ones in [1,6].The neighborhood  alternatively can be replaced by any accumulation point in the lower half complex plane or anywhere away from the poles of   (, , ), and the source point is fixed on .The advantage in this paper is that the argument does not rely on the Blaschke product of certain zero set.Theorem 3. The total wave field   (, , ) generated by potential   each has at most a finite number of zeros for each fixed (, ).If  1 (, , ) and  2 (, , ) share the same zero set for each fixed (, ), say, { 1 (),  2 (), . . .,  () ()}, then For problem PISP 2, we will demonstrate later that the total wave (, , ) behaves asymptotically as an exponential function for large  in the complex plane.We seek to apply the identity theorems in complex analysis to carry on the arguments.For PISP 1, the oscillation frequency of the scattered wave field is connected to the spectral invariant in the form of the Radon transform in analysis.In [1], it is shown that if (, , ) is provided for  ≫ 0, the Schrödinger potential can be reconstructed from the inverse Radon transform of ∫ (,) ().In particular, the inverse existence and uniqueness on the potential  are proved.In Theorem 1, we prove uniqueness by the information provided in a finite neighborhood of frequencies, and the existence follows accordingly.The proof of Theorem 1 is again the application of the spectral invariant ∫ (,) ().
In the last section, we discuss the inverse problem when only the phase information of the scattered wave fields is provided.It has been discussed in [14,15].
In the real-world applications, the measured data is taken in real-valued frequencies.In Appendix, we discuss the inverse uniqueness when provided with only the measured data on the real axis.The mechanism is due to the Nevanlinna-Levin type of integral representation theorem.

Preliminaries
The fundamental element is the asymptotics [ in which (, ) is the straight line connecting  and , and  is the arc length.We extend the property into the lower half complex plane in the following lemma for the sake of a complex analysis [16].
For far-field behavior, we have which holds uniformly for all x fl /||,  ∈ R 3 , and  ∞ (x; , ) is known as the scattering amplitude in the literature [8,10,18,19].In this paper, we adopt the convention that  ∞ (x; , ) is defined analytically in I ≤ 0 and extended meromorphically from I ≤ 0 to C. To avoid the poles of  ∞ (x; , ), we analyze the problem in I ≤ 0.
Lemma 5.The scattered wave field   (, , ) in ( 17) is defined meromorphically in C with poles in I > 0, except for a finite number of purely imaginary 's that  2 are the negative eigenvalues of (1).In particular, the poles of  ∞ (x; , ) are located as the mirror images of its zeros to the real axis.
Proof.This is well-known in scattering theory.Let us refer to [18][19][20][21] and in particular a few brief comments on the analytic structure of the scattering matrix in [20,Introduction].There are only finitely many poles located in the lower half complex plane.

A Proof of Theorem 1
Let    (, , ) be the scattered wave field induced by the Schrödinger potential   (), and We start with the assumption in Theorem 1; that is, For each fixed I and  ̸ = ,  ∈ , the function is real-analytic in R by avoiding the poles mentioned in Lemma 5. Hence, we extend the identity to a strip  containing 0 + R. We write in which the constant  2 is specified in (13) and  = () > 0.
Moreover, for each fixed R and  ̸ = ,  ∈ ,   (, R + I, ) is analytic in I.Thus, the identity (19) holds in (22) with possibly finite exception of poles.From ( 19) and ( 22), we deduce that which is purely imaginary.From the Cauchy-Riemann equation in the theory of complex variable, we deduce that ln{ We compare ( 25) and ( 26) for large  to deduce that for each fixed pair (, ) under the assumption in Theorem 1.
For each line segment (, ), we consider the inverse Radon transform [1,4,5].For any number  ∈ R, we consider the following sets: where   = √  2 −  2 is the radius of the circle   , − <  < .Let us try to parametrize (, ) in the setting of the Radon transform [22] on each   .

Phase Only Problem
The phase information of the scattered wave field or the Fourier transform plays a role in inverse problem [14,15,26].
In [15], two images are Fourier-transformed by swapping their phases which explains the importance of the phase information for image recovery.   2 () . (43) We repeat the inverse Radon transform argument in previous sections to conclude that  1 ≡    2 .