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

Authors: Lung-Hui Chen
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

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 some micro/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.


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## Body

## 1. Introduction

In this paper, we study the scattering problem of the Schrödinger equation given by(1)Δxux,k,y+k2ux,k,y-qxux,k,y=-δx-y,x∈R3,Ik≤0;ux,k,y=u0x,k,y+usx,k,y;limr→∞r∂ux,k,y∂r-ikux,k,y=0,in which the nonzero q(x) is the Schrödinger potential, y∈R3 is the source point, r≔|x-y|, and(2)u0x,k,y≔exp⁡-ikx-y4πx-y.The perturbation is defined inside(3)Ω≔x∈R3∣x<B,B>0with boundary S. In this paper, we assume(4)qx∈C4R3;R;qx≥0,∀x∈Ω;qx=0,∀x∈R3∖Ω.It is shown in Klibanov and Romanov [1] that problem (1) has a solution us(x,k,y) in C4(|x-y|≥η), for all η>0. The phaseless inverse scattering problem (PISP) [1–6] is to find the Schrödinger potential q(x) if the information is given or partially given on the following types of scattering data measured in far-fields:(5)PISP1:fx,k,y≔usx,k,y2,x,k,y∈S×C×S;(6)PISP2:gx,k,y≔ux,k,y2,x,k,y∈S×C×S.We call the inverse problem given (5) the phaseless inverse scattering problem of type 1 and (6) type 2. This is a long-standing open problem in the inverse scattering theory of quantum mechanics [7–11], and the breakthrough is made in a series of papers by Klibanov and Romanov [1–6, 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.Theorem 1.
Letfj(x,k,y) be the square modulus of the complex-valued scattered wave field generated by Schrödinger potential qj, j=1,2. If f1(x,k,y)≡f2(x,k,y) for all (x,k,y)∈S×U×S in a neighborhood U intersected with real axis; then q1≡q2.Theorem 2.
Letgj(x,k,y) be the square modulus of the complex-valued total scattered wave field generated by potential qj, j=1,2. If g1(x,k,y)≡g2(x,k,y) for all (x,k,y)∈S×U×S in a neighborhood U intersected with real axis; then q1≡q2.Here we provide similar results to the ones in [1, 6]. The neighborhood U alternatively can be replaced by any accumulation point in the lower half complex plane or anywhere away from the poles of us(x,k,y), and the source point is fixed on S. 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 fielduj(x,k,y) generated by potential qj each has at most a finite number of zeros for each fixed (x,y). If u1(x,k,y) and u2(x,k,y) share the same zero set for each fixed (x,y), say, {k1(x),k2(x),…,km(x)(x)}, then q1≡q2.For problem PISP 2, we will demonstrate later that the total waveu(x,k,y) behaves asymptotically as an exponential function for large k 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 f(x,k,y) is provided for k≫0, the Schrödinger potential can be reconstructed from the inverse Radon transform of ∫L(x,y)‍q(ξ)dσ. In particular, the inverse existence and uniqueness on the potential q 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 ∫L(x,y)‍q(ξ)dσ.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.


## 2. Preliminaries

The fundamental element is the asymptotics [1, Theorem1](7)usx,k,y=iexp⁡-ikx-y8πx-yk∫Lx,yqξdσ+O1k,x≠y,k⟶∞,in which L(x,y) is the straight line connecting x and y, and dσ 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].Lemma 4 (Klibanov and Romanov).
There exists some constantC>0 such that the following asymptotics holds in Ik<C(8)usx,k,y=iexp⁡-ikx-y8πx-yk∫Lx,yqξdσ+O1k,x≠y.Proof.
We review the fundamental solution of the hyperbolic equation [1, Sec.3](9)wtt=Δxw-qxw+4πδx-y,t,x,t∈R4;wt<0=0.In particular,(10)wx,t,y=w0x,t,y+w~x,t,yHt-x-y,(11)w0x,t,y=δt-x-yx-y,in which H(t) is the Heaviside function and w~(x,t,y) has the short-time behavior(12)limt→x-y+w~x,t,y=-12x-y∫Lx,yqξdσ.Moreover, the solution of (9) satisfies the following properties. There exist constants C1>0 and C2>0 depending on (y,q,R) such that(13)∂tkwx,t,y≤C1e-C2t,∀t≥t0,∀x∈x∈R3∣x-y<R,k=0,1,2,for any R>0.
Let(14)Vx,k,y≔14π∫0∞wx,t,ye-iktdt,x≠y,k∈0i+R.From the theory of [1, 5, 17], we note that u(x,k,y)=V(x,k,y) is the solution of (1). Thus, we deduce from (1), (10), (12), (13), and (14) that(15)usx,k,y=14π∫x-y∞w~x,t,ye-iktdt=-iexp⁡-ikx-yk-18πx-y∫Lx,yqξdσ-i4πk∂tw~x,x-y+,y-14πk2∫x-y∞∂t2w~x,t,ye-iktdt,in which the last term could be of exponential growth for Ik≫C2>0. For fixed x≠y and fixed Ik<C2, we deduce from the Riemann-Lebesgue lemma that(16)∫x-y∞∂t2w~x,t,ye-iktdt=∫x-y∞e-iRkt∂t2w~x,t,yeIktdt=o1,asRk⟶±∞.The lemma is thus proven.For far-field behavior, we have(17)usx,k,y=eikxxu∞x^;y,k+O1x3/2,which holds uniformly for all x^≔x/|x|, x∈R3, and u∞(x^;y,k) is known as the scattering amplitude in the literature [8, 10, 18, 19]. In this paper, we adopt the convention that u∞(x^;y,k) is defined analytically in Ik≤0 and extended meromorphically from Ik≤0 to C. To avoid the poles of u∞(x^;y,k), we analyze the problem in Ik≤0.Lemma 5.
The scattered wave fieldus(x,k,y) in (17) is defined meromorphically in C with poles in Ik>0, except for a finite number of purely imaginary k’s that k2 are the negative eigenvalues of (1). In particular, the poles of u∞(x^;y,k) 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 [18–21] 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.


## 3. A Proof of Theorem1

Letusj(x,k,y) be the scattered wave field induced by the Schrödinger potential qj(x), and(18)fjx,k,y≔usjx,k,y2.We start with the assumption in Theorem 1; that is,(19)f1x,k,y=f2x,k,y,x,k,y∈S×U×S.For each fixed Ik and x≠y, k∈U, the function(20)fjx,k,y=Rusjx,k,y2+Iusjx,k,y2=Rusjx,Rk+iIk,y2+Iusjx,Rk+iIk,y2is real-analytic in Rk by avoiding the poles mentioned in Lemma 5. Hence, we extend the identity to a strip H containing 0i+R. We write(21)H≔k∈C∣-∞<Rk<∞,-δ<Ik<δ<C2,in which the constant C2 is specified in (13) and δ=δ(U)>0. Moreover, for each fixed Rk and x≠y, k∈H, fj(x,Rk+iIk,y) is analytic in Ik. Thus, the identity (19) holds in(22)C-≔k∈C∣-∞<Rk<∞,-∞<Ik<C2,with possibly finite exception of poles. From (19) and (22), we deduce that(23)us1x,k,y=us2x,k,y.Accordingly, us1(x,k,y) and us2(x,k,y) have identical zero set in C-, and therein us1(x,k,y)/us2(x,k,y) is analytic. Thus,(24)ln⁡us1x,k,yus2x,k,y=ln⁡us1x,k,yus2x,k,y+iarg⁡us1x,k,yus2x,k,y=iarg⁡us1x,k,yus2x,k,y,which is purely imaginary. From the Cauchy-Riemann equation in the theory of complex variable, we deduce that ln{us1(x,k,y)/us2(x,k,y)} is a constant, say, iγ∈iR. Therefore,(25)us1x,k,yus2x,k,y=eiγ.From Lemma 4, we consider(26)us1x,k,yus2x,k,y=iexp⁡-ikx-y/8πx-yk∫Lx,yq1ξdσ+O1/kiexp⁡-ikx-y/8πx-yk∫Lx,yq2ξdσ+O1/k=∫Lx,yq1ξdσ+O1/k∫Lx,yq2ξdσ+O1/k≳0,k∈C-.We compare (25) and (26) for large k to deduce that(27)∫Lx,yq1ξdσ=∫Lx,yq2ξdσ,for each fixed pair (x,y) under the assumption in Theorem 1.For each line segmentL(x,y), we consider the inverse Radon transform [1, 4, 5]. For any number a∈R, we consider the following sets:(28)Pa≔x=x1,x2,x3∈R3∣x3=a;Sa≔S∩Pa,where Ba=B2-a2 is the radius of the circle Sa, -B<a<B. Let us try to parametrize L(x,y) in the setting of the Radon transform [22] on each Pa.We consider the outer unit normal vectorn to the line L(x,y) lying on the plane Pa. For each fixed x3=a, we denote α∈(0,2π] as the angle between normal vector n and x1-axis. Then n=n(α)=(cos⁡α,sin⁡α,0). For the third coordinate, we set s∈(-Ba,Ba) to be the signed distance of L(x,y) to (0,0,a). It is clear that there is one-to-one correspondence between pairs (x,y) and (n(α),s). That is, we can identify (x,y) and (n(α),s) by each other. Therefore, we can write for each fixed a∈(-B,B)(29)Lx,y=ya≔y1,y2,a∈R3∣ya·nα=s.Thus, we derive the 2-dimensional Radon transform(30)∫Lx,yqjξdσ=∫ya·nα=sqjyadσ,j=1,2.From (27) and the inverse uniqueness of the Radon transform [22], we deduce that q1(ya)≡q2(ya) on each fixed Pa. Hence, we conclude that(31)q1x≡q2x.


## 4. A Proof of Theorem2

We start the total wave fielduj(x,k,y)=u0(x,k,y)+usj(x,k,y), j=1,2. Therefore,(32)ujx,k,y=exp-ikx-y4πx-y+iexp⁡-ikx-y8πx-yk∫Lx,yqjξdσ+O1k=exp-ikx-y4πx-y1+i2k∫Lx,yqjξdσ+O1k2=exp-ikx-y4πx-y1+O1k,k∈C-.Hence,(33)ujx,k,y-exp-ikx-y4πx-y<exp-ikx-y4πx-yO1k≲exp-ikx-y4πx-y,for |k|≫0 in C-. From Rouché’s theorem in complex analysis, uj(x,k,y) and exp{-ik|x-y|}/4π|x-y| have identical number of zeros in regions away from the origin. Hence, uj(x,k,y) has only finitely many zeros near the origin. For |k|≫0, we apply the assumption in Theorem 2 to deduce that(34)ln⁡u1x,k,yu2x,k,y=ln⁡u1x,k,yu2x,k,y+iarg⁡u1x,k,yu2x,k,y=iarg⁡u1x,k,yu2x,k,y,which is again a purely imaginary function that extends to hold in |k|≫0 in C-. Hence, we conclude that u1(x,k,y)/u2(x,k,y)=eiδ for some real constant δ. Now we apply (32) to obtain(35)eiδ=1+i/2k∫Lx,yq1ξdσ+O1/k21+i/2k∫Lx,yq2ξdσ+O1/k2,k∈C-.We deduce that δ=0, and then ∫L(x,y)‍q1(ξ)dσ=∫L(x,y)‍q2(ξ)dσ. For each fixed line segment L(x,y) and Pa, we apply the inverse Radon transform constructed in previous section and then prove q1≡q2.


## 5. A Proof of Theorem3

The total wave fieldsu1(x,k,y) and u1(x,k,y) have only finite number of zeros due to (32), from which we deduce that(36)u1x,k,yu2x,k,y=1+i/2k∫Lx,yq1ξdσ+O1/k21+i/2k∫Lx,yq2ξdσ+O1/k2=1+O1k,k∈C-.For each fixed (x,y), we denote(37)Pk≔k-k1k-k2⋯k-km.Accordingly, we consider the analytic function(38)u1x,k,yu2x,k,y=u1x,k,y/Pku2x,k,y/Pk=1+O1k,k∈C-.We apply the Phragmén-Lindelöf Theorem [23–25] to conclude that u1(x,k,y)/u2(x,k,y)≡1 in Ik≤0. We repeat the argument in previous section to deduce that ∫L(x,y)‍q1(ξ)dσ=∫L(x,y)‍q2(ξ)dσ for the fixed L(x,y). The theorem is thus proven.


## 6. 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.Let us put the scattered wave fields in polar form(39)usx,k,y≔usx,k,yexp⁡iarg⁡ϕx,k,y;ux,k,y≔ux,k,yexp⁡iarg⁡φx,k,y,x,k,y∈S×C×S.In this section, we consider the inverse problems involving with phase information ϕ(x,k,y) and φ(x,k,y).Theorem 6.
Letϕj(x,k,y) be the phase information regarding the scatter wave field generated by qj, j=1,2. If ϕ1(x,k,y)≡ϕ2(x,k,y) for all (x,k,y)∈S×U×S in a neighborhood U intersected with real axis, then q1≡eγq2, for some real constant γ.Proof.
We begin with the logarithmic function(40)ln⁡us1x,k,yus2x,k,y=ln⁡us1x,k,yus2x,k,y+iarg⁡ϕ1x,k,yϕ2x,k,y=ln⁡us1x,k,yus2x,k,y+iarg⁡ϕ1x,k,y-iarg⁡ϕ2x,k,y=ln⁡us1x,k,yus2x,k,y,by theorem assumption. We note that it is purelyreal in U. Moreover, ln{us1(x,k,y)/us2(x,k,y)} is meromorphic in C by (17) if us2(x,k,y) is not identically zero. We deduce from the Cauchy-Riemann equation in the theory of complex variable that ln{us1(x,k,y)/us2(x,k,y)} is a real constant, say, γ outside the possible poles in C. Thus, we deduce that, outside some possible poles,(41)us1x,k,yus2x,k,y=eγ.Moreover, we apply Lemma 4 again to obtain(42)us1x,k,yus2x,k,y=∫Lx,yq1ξdσ+O1/k∫Lx,yq2ξdσ+O1/k=eγ,Ik≤0.Hence, for each L(x,y),(43)∫Lx,yq1ξdσ=eγ∫Lx,yq2ξdσ=∫Lx,yeγq2ξdσ.We repeat the inverse Radon transform argument in previous sections to conclude that q1≡eγq2.In this case, the uniqueness is not available. We can carry out similar analysis to study total wave fieldu(x,k,y).Theorem 7.
Letϕj(x,k,y) be the phase information regarding the scattered total wave field generated by qj, j=1,2. If ϕ1(x,k,y)≡ϕ2(x,k,y) for all (x,k,y)∈S×U×S in some neighborhood U intersected with real axis, then q1≡q2.Proof.
The proof is similar to Theorem6. We repeat the proof of (40), (41), and (36) to deduce(44)eγ=u1x,k,yu2x,k,y=1+i/2k∫Lx,yq1ξdσ+O1/k21+i/2k∫Lx,yq2ξdσ+O1/k2=1+O1k,k∈C-.Here, we find γ=0. Then, u1(x,k,y) and u2(x,k,y) have the same zero set. It proves the theorem by repeating the argument in proving Theorem 3.



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