AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi Publishing Corporation 10.1155/2016/6031523 6031523 Research Article Remarks on the Phaseless Inverse Uniqueness of a Three-Dimensional Schrödinger Scattering Problem http://orcid.org/0000-0001-9535-7747 Chen Lung-Hui 1 Weder Ricardo Department of Mathematics National Chung Cheng University 168 University Rd. Min-Hsiung Chia-Yi County 621 Taiwan ccu.edu.tw 2016 13112016 2016 31 08 2016 19 10 2016 13112016 2016 Copyright © 2016 Lung-Hui Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Ministry of Science and Technology, Taiwan NSC 99-2115-M-194-004
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,xR3,Ik0;ux,k,y=u0x,k,y+usx,k,y;limrrux,k,yr-ikux,k,y=0,in which the nonzero q(x) is the Schrödinger potential, yR3 is the source point, r|x-y|, and(2)u0x,k,yexp-ikx-y4πx-y.The perturbation is defined inside(3)ΩxR3x<B,B>0with boundary S. In this paper, we assume(4)qxC4R3;R;qx0,xΩ;qx=0,xR3Ω.It is shown in Klibanov and Romanov  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)  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,yusx,k,y2,x,k,yS×C×S;(6)PISP2:gx,k,yux,k,y2,x,k,yS×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 , and the breakthrough is made in a series of papers by Klibanov and Romanov [16, 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 . 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.

Let fj(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 q1q2.

Theorem 2.

Let gj(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 q1q2.

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 field uj(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 q1q2.

For problem PISP 2, we will demonstrate later that the total wave u(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 , it is shown that if f(x,k,y) is provided for k0, 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, Theorem  1](7)usx,k,y=iexp-ikx-y8πx-ykLx,yqξdσ+O1k,xy,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 .

Lemma 4 (Klibanov and Romanov).

There exists some constant C>0 such that the following asymptotics holds in Ik<C(8)usx,k,y=iexp-ikx-y8πx-ykLx,yqξdσ+O1k,xy.

Proof.

We review the fundamental solution of the hyperbolic equation [1, Sec.  3](9)wtt=Δxw-qxw+4πδx-y,t,x,tR4;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)limtx-y+w~x,t,y=-12x-yLx,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,yC1e-C2t,tt0,xxR3x-y<R,k=0,1,2,for any R>0.

Let(14)Vx,k,y14π0wx,t,ye-iktdt,xy,k0i+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=14πx-yw~x,t,ye-iktdt=-iexp-ikx-yk-18πx-yLx,yqξdσ-i4πktw~x,x-y+,y-14πk2x-yt2w~x,t,ye-iktdt,in which the last term could be of exponential growth for IkC2>0. For fixed xy and fixed Ik<C2, we deduce from the Riemann-Lebesgue lemma that(16)x-yt2w~x,t,ye-iktdt=x-ye-iRktt2w~x,t,yeIktdt=o1,asRk±.The lemma is thus proven.

For far-field behavior, we have(17)usx,k,y=eikxxux^;y,k+O1x3/2,which holds uniformly for all x^x/|x|, xR3, 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 Ik0 and extended meromorphically from Ik0 to C. To avoid the poles of u(x^;y,k), we analyze the problem in Ik0.

Lemma 5.

The scattered wave field us(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  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 Theorem <xref ref-type="statement" rid="thm1.1">1</xref>

Let usj(x,k,y) be the scattered wave field induced by the Schrödinger potential qj(x), and(18)fjx,k,yusjx,k,y2.We start with the assumption in Theorem 1; that is,(19)f1x,k,y=f2x,k,y,x,k,yS×U×S.For each fixed Ik and xy, kU, 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)HkC-<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 xy, kH, fj(x,Rk+iIk,y) is analytic in Ik. Thus, the identity (19) holds in(22)C-kC-<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)lnus1x,k,yus2x,k,y=lnus1x,k,yus2x,k,y+iargus1x,k,yus2x,k,y=iargus1x,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, iγiR. Therefore,(25)us1x,k,yus2x,k,y=eiγ.From Lemma 4, we consider(26)us1x,k,yus2x,k,y=iexp-ikx-y/8πx-ykLx,yq1ξdσ+O1/kiexp-ikx-y/8πx-ykLx,yq2ξdσ+O1/k=Lx,yq1ξdσ+O1/kLx,yq2ξdσ+O1/k0,kC-.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 segment L(x,y), we consider the inverse Radon transform [1, 4, 5]. For any number aR, we consider the following sets:(28)Pax=x1,x2,x3R3x3=a;SaSPa,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  on each Pa.

We consider the outer unit normal vector n 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)=(cosα,sinα,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=yay1,y2,aR3ya·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 , we deduce that q1(ya)q2(ya) on each fixed Pa. Hence, we conclude that(31)q1xq2x.

4. A Proof of Theorem <xref ref-type="statement" rid="thm1.2">2</xref>

We start the total wave field uj(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-ykLx,yqjξdσ+O1k=exp-ikx-y4πx-y1+i2kLx,yqjξdσ+O1k2=exp-ikx-y4πx-y1+O1k,kC-.Hence,(33)ujx,k,y-exp-ikx-y4πx-y<exp-ikx-y4πx-yO1kexp-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)lnu1x,k,yu2x,k,y=lnu1x,k,yu2x,k,y+iargu1x,k,yu2x,k,y=iargu1x,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/2kLx,yq1ξdσ+O1/k21+i/2kLx,yq2ξdσ+O1/k2,kC-.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 q1q2.

5. A Proof of Theorem <xref ref-type="statement" rid="thm1.3">3</xref>

The total wave fields u1(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/2kLx,yq1ξdσ+O1/k21+i/2kLx,yq2ξdσ+O1/k2=1+O1k,kC-.For each fixed (x,y), we denote(37)Pkk-k1k-k2k-km.Accordingly, we consider the analytic function(38)u1x,k,yu2x,k,y=u1x,k,y/Pku2x,k,y/Pk=1+O1k,kC-.We apply the Phragmén-Lindelöf Theorem  to conclude that u1(x,k,y)/u2(x,k,y)1 in Ik0. 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 , 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,yusx,k,yexpiargϕx,k,y;ux,k,yux,k,yexpiargφx,k,y,x,k,yS×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.

Let ϕj(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 q1eγq2, for some real constant γ.

Proof.

We begin with the logarithmic function(40)lnus1x,k,yus2x,k,y=lnus1x,k,yus2x,k,y+iargϕ1x,k,yϕ2x,k,y=lnus1x,k,yus2x,k,y+iargϕ1x,k,y-iargϕ2x,k,y=lnus1x,k,yus2x,k,y,by theorem assumption. We note that it is purely real 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/kLx,yq2ξdσ+O1/k=eγ,Ik0.Hence, for each L(x,y),(43)Lx,yq1ξdσ=eγLx,yq2ξdσ=Lx,yeγq2ξdσ.We repeat the inverse Radon transform argument in previous sections to conclude that q1eγq2.

In this case, the uniqueness is not available. We can carry out similar analysis to study total wave field u(x,k,y).

Theorem 7.

Let ϕj(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 q1q2.

Proof.

The proof is similar to Theorem 6. We repeat the proof of (40), (41), and (36) to deduce(44)eγ=u1x,k,yu2x,k,y=1+i/2kLx,yq1ξdσ+O1/k21+i/2kLx,yq2ξdσ+O1/k2=1+O1k,kC-.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.

Appendix Complex Analysis and the Nevanlinna-Levin Theorem

In the real-world applications, the scattered data are measured in real-valued frequencies. The assumption we have adopted so far is focused on the neighborhoods centered at real axis. In this appendix, we use complex analysis to relax the assumptions in Theorems 1, 2, 3, 6, and 7. The Fourier transforms (14) and (15) behave like exponential functions on many aspects. The integral representation theorem plays a role.

Definition A.1.

Let f(z) be an entire function. Let Mf(r)max|z|=r|f(z)|. An entire function of f(z) is said to be a function of finite order if there exists a positive constant k such that the inequality(A.1)Mfr<erkis valid for all sufficiently large values of r. The greatest lower bound of such numbers k is called the order of the entire function f(z). By the type σ of an entire function f(z) of order ρ, we mean the greatest lower bound of positive number A for which asymptotically we have(A.2)Mfr<eArρ.That is,(A.3)σlimsuprlnMfrrρ.If 0<σ<, then we say f(z) is of normal type or mean type.

Definition A.2.

Let f(z) be an integral function of finite order ρ in the angle [θ1,θ2]. We call the following quantity as the indicator function of the function f(z):(A.4)hfθlimrlnfreiθrρ,θ1θθ2.

The type of a function is connected to the maximal value of the indicator function. We state it as the following lemma.

Lemma A.3 (Ja Levin [<xref ref-type="bibr" rid="B20">24</xref>, p. 72]).

The maximum value of the indicator hf(θ) of the function f(z) on the interval αθβ is equal to the type σ of this function inside the angle αargzβ.

Lemma A.4.

The analytic function usj(x,k,y), j=1,2, is bounded in the lower half complex plane.

Proof.

We recall (8); that is,(A.5)usx,k,y=iexp-ikx-y8πx-ykLx,yqξdσ+O1k,xy,Ik0.Hence,(A.6)usx,k,y=expIkx-yO1k,Ik0,which is bounded for negative Ik.

We use the integral representation theorem due to Nevanlinna and Levin [23, 24, 27].

Definition A.5.

Let f(z) be an analytic function in the upper half-plane. We say u(z) is a harmonic majorant of log|f(z)| if log|f(z)|u(z), and u(z) is a harmonic function in the upper half-plane.

From Lemma A.4, the function f(z) is bounded in the lower half-plane. Thus, log|f(z)| is bounded by a constant which is trivially harmonic.

Theorem A.6.

Let f(z) be an analytic function in {Iz>0}, and let the function log|f(z)| have a positive harmonic majorant in {Iz>0}. Then(A.7)logfz=k=1logz-akz-a¯k+yπ-logftt-z2dt+σy,where {ak} are the zeros of f(z) in {Iz>0}.

Proof.

We refer the proof to [27, p. 104] and the remark therein. We also refer more connection of the integral representation theorem to the other analytic properties of the function [27, p. 115, p. 116].

Theorem A.7.

If f1(x,k,y)=f2(x,k,y) for k in a neighborhood IR+0i with (x,y) fixed, then |us1(x,k,y)|=|us2(x,k,y)| holds in C-.

Proof.

Given |us1(x,k,y)|2=|us2(x,k,y)|2 for all kI, the identity for the real-analytic functions extends to hold in real axis. Hence, we have(A.8)us1x,k,y=us2x,k,y,kR+0i.For fixed (x,y), the type of usj(x,k,y), j=1,2, is zero in Ik0 by following Definitions A.1 and A.2, Lemmas A.3 and A.4. In (A.7), let us set σ=1. Therefore, we apply Theorem A.6 in Ik0 to deduce(A.9)logus1x,k,y-logus2x,k,y=n=1mlogk-ank-a¯n-n=1mlogk-ank-a¯n,Ik0,where {an} and {an} are the zeros of us1(x,k,y) and us2(x,k,y), respectively, in the lower half-plane. That is,(A.10)us1x,k,yus2x,k,y=n=1mk-an/k-a¯nn=1mk-an/k-a¯n,Ik0,which is the absolute value of a rational function in k.

From (A.8) and (A.10), we deduce that(A.11)us1x,k,yus2x,k,y1.Moreover, the identity extends to be valid up to Ik<C2 discussed as in (22). The theorem is thus proven.

Competing Interests

The author declares that there is no conflict of interest regarding the publication of this manuscript.

Acknowledgments

This author’s work is supported by Ministry of Science and Technology, NSC 99-2115-M-194-004.

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