Wave Support Theorem and Inverse Resonant Uniqueness on the Line
WWave Support Theoremand Inverse Resonant Uniqueness on the Line
Lung-Hui Chen June 5, 2020
Abstract
In the paper, we experimentally study the inverse problem with the resonant scattering determi-nant. We analyze the structure of characteristics of perturbed linear waves. Assuming there is thecommon part of potential perturbation propagating along the same strips, we estimate the commonpart of the perturbed wave, and its Fourier transform. We deduce the partial inverse uniquenessfrom the Nevanlinna type of representation theorem.MSC: 34B24/35P25/35R30.Keywords: scattering; resonance; Schr¨odinger equation; inverse problem; Nevanlinna theorem.
Let us consider the Schr¨odinger equation − d dx + V ( x ) , x ∈ R , V ( x ) ∈ L comp ([ a, b ]) , a < < b, | a | (cid:28) | b | , (1.1)where we assume the potential is effectively support on [ a, b ], that is, [ a, b ] is the minimal convex hullthat contains the support of V j . The scattering matrix is of the form S ( k ) = ik ˆ X ( k ) ˆ Y ( k )ˆ X ( k )ˆ Y ( − k )ˆ X ( k ) ik ˆ X ( k ) . (1.2)The scattering matrix S ( k ) is meromorphic in C , and its poles in {(cid:61) k > } are the square roots of L -eigenvalues of (1.1). In this paper, we understand ˆ X ( k ) through the one-dimensional wave equation (cid:40) (cid:0) D x − D y + V ( x ) (cid:1) A ± ( x, y ) = 0; A ± ( x, y ) = δ ( x − y ) , ± x (cid:29) , (1.3)where D y A − ( x, y ) = X ( y − x ) + Y ( y + x ). In particular [9, p. 727], X ( x ) − δ (cid:48) ( x ) + (cid:82) V ( t ) dt δ ( x ) ∈ L ( R ) ∩ L ∞ ( R ); (1.4) Y ( y ) − V ( y/ ∈ L ( R ) ∩ L ∞ ( R ) . (1.5)Thus, A ± ( x, y ) satisfies the wave equation with x taking the place of time (this choice is dictated by theforcing condition imposed) in (1.3). The uniqueness part follows from the energy estimates of the waveequation [2, 9, 10, 12]. General Education Center, Ming Chi University of Technology, New Taipei City, 24301, Taiwan. Email:[email protected]. a r X i v : . [ m a t h . SP ] J un n this paper, we consider the complex analysis of entire function ˆ X ( k ) and ˆ Y ( k ) which are representedin form of ˆ X ( k ) = (cid:90) − b − a ) X ( x ) e − ikx dx ; (1.6)ˆ Y ( k ) = (cid:90) b a Y ( y ) e − iky dy, k ∈ C , (1.7)that the unitary identity holds in C :ˆ X ( k ) ˆ X ( − k ) = k + ˆ Y ( k ) ˆ Y ( − k ) , (1.8)More importantly, we consider experimentallydet S ( k ) := − ˆ X ( − k )ˆ X ( k ) . (1.9)as scattering data in this paper inspired its simpler analytic structure.We define for potential V j , j = 1 ,
2, 0 < r (cid:28) X j ( k ) := F{ X j χ [2 a, ( y − b ) } ( k ); (1.10)ˆ X j ( k ) := F{ X j χ [2 a − r, a ] ( y − b ) } ( k ); (1.11)ˆ X j ( k ) := F{ X j χ [2 a − b, a − r ] ( y − b ) } ( k ); (1.12)ˆ Y j ( k ) := F{ X j χ [2 a, ( y + b ) } ( k ); (1.13)ˆ Y j ( k ) := F{ X j χ [0 , r ] ( y + b ) } ( k ); (1.14)ˆ Y j ( k ) := F{ X j χ [2 r, b ] ( y + b ) } ( k ) , (1.15)in which χ [2 a, ( x ) is the characteristic function defined on [2 a, D y A j − ( x, y ) is illustrated in the shaded areas in Figure 1. Most importantly, X j χ [2 a − r, a ] ( y − b )in not in the domain of influence of V j χ [ a,
0] when time variable x = b . The function D y A j − ( x, y ) = X ( y − x ) + Y ( y + x ) satisfies the wave equation for x ≥ b . For x ≤ a , we have D y A j − ( x, y ) = − δ (cid:48) ( x − y ).If b = 0, we firstly see the support of X j χ [2 a, ( y ). For b (cid:29)
0, we find the red triangle in the center ofdiagram that shows the support of the wave solution that is not affected by the V j χ [ a, V j χ [0 , b ]. We refer more detailed analysis construction and characteristics analysis to Figure1. Let det S j ( k ) := − ˆ X j ( − k )ˆ X j ( k ) , j = 1 , , be the corresponding scattering determinant of of potential V j . Theorem 1.1. If V ( x ) ≡ V ( x ) on non-empty [0 , b ] ⊂ [ a, b ] , and det S ( k ) = det S ( k ) , then V ( x ) ≡ V ( x ) on [ a, b ] . In literature, when there are no bound states, the potential is determined by the reflection coefficientsby Faddeev’s theory [3, 4]. In inverse resonance problem, we consider to determine the potential V fromthe resonances of (1.1) which includes the square root of L -eigenvalues. The inverse resonance problemof Schr¨odinger operator on the half line has been studied in [5, 6, 12]. In the half line case, the uniquerecovery of the potential from the eigenvalues and resonances is justified in [5]. However, in the fullline case, the inverse resonance problems mainly remained open for a long time. It is known that thepotential cannot be solely determined by the eigenvalues and resonances. Specifically, Zworski [12] provedthe uniqueness theorem for the symmetric potentials along with certain isopolar results. Furthermore,Korotyaev [5, 6] applied the value distribution theory in complex analysis to prove that all eigenvaluesand resonances, and a signed sequence can uniquely determine the potential V .2 y x=b0x=a support of V=[a,b] (b,b)(b,2a-b)(a,a) (0,2a) support of X=[2a-2b,0]support of X support of Y support of Y D y A j − ( x , y ) = − δ ′ ( x − y ) Figure 1: characteristics of linear waves
Lemma 2.1. If F ( x ) is of bounded variation on ( −∞ , ∞ ) , then F ( x ) is constant except on some finiteinterval if and only if f ( z ) = (cid:82) ∞−∞ e izt dF ( t ) is an entire functional of exponential type; and if ( a, b ) isthe smallest interval outside which F ( x ) is a constant, then a = h f ( − π ) and b = h f ( π ) .Proof. We refer the proof to Boas [1, p. 109].Here, we note that the indicator diagram of ˆ X j ( k ) is the line set i [ − b − a ) ,
0] on the imaginaryaxis, and its length is | a | + | b | . Lemma 2.2.
The length of indicator diagram of ˆ X j ( k ) is b − a ) .Proof. We use Lemma 2.1 and (1.6) to conclude the result.
Lemma 2.3. ˆ X j ( k ) has indicator function | a || sin θ | ; ˆ X j ( k ) has indicator function | r || sin θ | ; ˆ X j ( k ) hasindicator function | b − r || sin θ | .Proof. It is straightforward by Lemma 2.1, (1.10), (1.11), and (1.12).
Lemma 2.4. ˆ X jj (cid:48) ( k ) has only finite zeros in C + , and infinitely many in C − .Proof. This is well-known in the literature, say, [2].
Lemma 2.5. If V ≡ V on [0 , b ] , then ˆ X ( k ) ≡ ˆ X ( k ) , and ˆ Y ( k ) ≡ ˆ Y ( k ) .Proof. Let us discuss the Figure 1. Let us first take b = 0, so the wave starting inside the triangle: (0 , a, a ), (0 , a ) propagate in parallel strips between the shaded area between the lines pointing at ( a, a )and (0 , a ) and the lines pointing at ( a, a ) and (0 , X j and Y j , j = 1 ,
2, are definedon the those two strips correspondingly. Now we consider x ∈ [0 , b ], if V ≡ V on [0 , b ], then the wavefunction Y ≡ Y are defined on the strips between the lines contain (0 ,
0) and ( b, b ). Similarly, the wavefunction X ≡ X are defined on the strips between the lines contain (0 , a ) and ( b, a − b ). Now wededuce from (1.10) that ˆ X ( k ) ≡ ˆ X ( k ). Similar arguments works for ˆ Y ( k ) ≡ ˆ Y ( k ).3 Proof of Theorem 1.1
We start with the assumption of Theorem 1.1det S ( k ) ≡ det S ( k ) , that is ˆ X ( − k )ˆ X ( k ) ≡ ˆ X ( − k )ˆ X ( k ) . (3.1)Using Lemma 2.4 and comparing the poles on both sides of (3.1), ˆ X ( k ) and ˆ X ( k ) have common zerosin C − , except finite ones in C + . If σ is a common zero of ˆ X ( k ) and ˆ X ( k ), thenˆ X ( σ ) + ˆ X ( σ ) = ˆ X ( σ ) + ˆ X ( σ ) = 0 , (3.2)by Lemma 2.5. Now we want to show thatˆ X ( k ) := ˆ X ( k ) + ˆ X ( k ) + ˆ X ( k ) ≡ ˆ X ( k ) + ˆ X ( k ) + ˆ X ( k ) =: ˆ X ( k ) . (3.3)That is, ˆ X ( k ) + ˆ X ( k ) ≡ ˆ X ( k ) + ˆ X ( k ) . (3.4)By Lemma 2.5, (3.3) and (3.4) should have same density of common zeros.Let us count the zero set of F ( k ) := ˆ X j ( k ) , j = 1 , , (3.5)and the zero set of G ( k ) := ˆ X ( k ) + ˆ X ( k ) − ˆ X ( k ) − ˆ X ( k ) . (3.6)Using (4.4) and (4.5) in Appendix, we deduce that h F ( θ ) = ( b − a ) | sin θ | ; (3.7) h G ( θ ) = max {| a | , | b − r |}| sin θ | = | b − r || sin θ | . (3.8)Using Theorem 4.5, F ( z ) has zero density ( b − a ) π , and G ( k ) has identical zero density b − rπ . That contra-dicts to Lemma 2.5. Then, we deduce thatˆ X ( k ) + ˆ X ( k ) ≡ ˆ X ( k ) + ˆ X ( k ) . Due to Lemma 2.5, we have ˆ X ( k ) ≡ ˆ X ( k ) . Using (1.8), we have ˆ X j ( k ) ˆ X j ( − k ) = k + ˆ Y j ( k ) ˆ Y j ( − k ) . (3.9)Thus, we obtain ˆ Y ( k ) ˆ Y ( − k ) ≡ ˆ Y ( k ) ˆ Y ( − k ) , k ∈ C . (3.10)Equivalently, | ˆ Y ( k ) | = | ˆ Y ( k ) | , k ∈ R , and we apply Theorem 4.8 to deduceˆ Y ( k ) ∞ (cid:89) n =1 − ka n − ka n = e iγ ˆ Y ( k ) ∞ (cid:89) n =1 − ka n − za n , (3.11)where { a jn } are the zeros of ˆ Y j ( k ) in C + . The Blaschke product ∞ (cid:89) n =1 − ka jn − ka jn
4s a function of zero type and of zero density of zeros. We refer the detail to [1]. Again, the zero densityof (3.11) is b − aπ . Thus, ˆ Y ( k ) and ˆ Y ( k ) have common zero of density b − aπ . That is,ˆ Y ( σ ) = ˆ Y ( σ ) , ∀ σ ∈ Σ , (3.12)in which Σ is a set of density b − aπ . That is,ˆ Y ( σ ) + ˆ Y ( σ ) + ˆ Y ( σ ) = ˆ Y ( σ ) + ˆ Y ( σ ) + ˆ Y ( σ ) , ∀ σ ∈ Σ . (3.13)That is, by Lemma 2.5, ˆ Y ( σ ) + ˆ Y ( σ ) = ˆ Y ( σ ) + ˆ Y ( σ ) , ∀ σ ∈ Σ , (3.14)which however could have a zero set of density b − a − rπ . Hence, ˆ Y ( k ) + ˆ Y ( k ) ≡ ˆ Y ( k ) + ˆ Y ( k ), and thenˆ Y ( k ) ≡ ˆ Y ( k ) . Therefore, we deduce that V and V have the same scattering matrix; S ( k ) ≡ S ( k ) . (3.15)Using Zworski [11, Proposition 8], which says that the potential function with compact support is deter-mined by the scattering matrix, we deduce that V ( x ) ≡ V ( x ) . (3.16)This proves the theorem. We review some results from complex analysis [1, 8].
Definition 4.1.
Let f ( z ) be an entire function. Let M f ( 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 suchthat the inequality M f ( r ) < e r k is valid for all sufficiently large values of r . The greatest lower bound of such numbers k is called theorder of the entire function f ( z ). By the type σ of an entire function f ( z ) of order ρ , we mean thegreatest lower bound of positive number A for which asymptotically we have M f ( r ) < e Ar ρ . That is, σ f := lim sup r →∞ ln M f ( r ) r ρ . If 0 < σ f < ∞ , then we say f ( z ) is of normal type or mean type. For σ f = 0, we say f ( z ) is of minimaltype.We refer the details to [8]. Definition 4.2.
Let f ( z ) be an integral function of finite order ρ in the angle [ θ , θ ]. We call thefollowing quantity as the indicator function of function f ( z ). h f ( θ ) := lim r →∞ ln | f ( re iθ ) | r ρ , θ ≤ θ ≤ θ . The type of a function is connected to the maximal value of indicator function.5 efinition 4.3.
The following quantity is called the width of the indicator diagram of entire function f : d = h f ( π h f ( − π . (4.1) Definition 4.4.
Let f ( z ) be an integral function of order 1, and let n ( f, α, β, r ) denote the number ofthe zeros of f ( z ) inside the angle [ α, β ] and | z | ≤ r . We define the density function as∆ f ( α, β ) := lim r →∞ n ( f, α, β, r ) r , and ∆ f ( β ) := ∆ f ( α , β ) , with some fixed α / ∈ E such that E is at most a countable set [1, 7, 8]. In particular, we denote thedensity function of f on the open right/left half complex plane as ∆ + f /∆ − f respectively. Similarly, wecan define the set density of a zero set S . Let n ( S, r ) be the number of the discrete elements of S in {| z | < r } . We define ∆ S := lim r →∞ n ( S, r ) r , Theorem 4.5 (Cartwright) . Let f be an entire function of exponential type with zero set { a k } . Weassume f satisfies one of the following conditions:the integral (cid:90) ∞−∞ ln + | f ( x ) | x dx exists . | f ( x ) | is bounded on the real axis . Then1. all of the zeros of the function f ( z ) , except possibly those of a set of zero density, lie inside arbitrarilysmall angles | arg z | < (cid:15) and | arg z − π | < (cid:15) , where the density ∆ f ( − (cid:15), (cid:15) ) = ∆ f ( π − (cid:15), π + (cid:15) ) = lim r →∞ n ( f, − (cid:15), (cid:15), r ) r = lim r →∞ n ( f, π − (cid:15), π + (cid:15), r ) r , (4.2) is equal to d π , where d is the width of the indicator diagram in (4.1). Furthermore, the limit δ = lim r →∞ δ ( r ) exists, where δ ( r ) := (cid:88) {| a k | 2. moreover, ∆ f ( (cid:15), π − (cid:15) ) = ∆ f ( π + (cid:15), − (cid:15) ) = 0; 3. the function f ( z ) can be represented in the form f ( z ) = cz m e iκz lim r →∞ (cid:89) {| a k | Let f , g be two entire functions. Then the following two inequalities hold. h fg ( θ ) ≤ h f ( θ ) + h g ( θ ) , if one limit exists ; (4.4) h f + g ( θ ) ≤ max θ { h f ( θ ) , h g ( θ ) } , (4.5) where the equality in (4.4) holds if one of the functions is of completely regular growth, and secondly theequality (4.5) holds if the indicator of the two summands are not equal at some θ . roof. We can find the details in [7]. Lemma 4.7. The Fourier transform ˆ X ( z ) as in (1.6) is of Cartwright class, and the function can berepresented in the form ˆ X ( z ) = cz m e iδz lim R →∞ (cid:89) | σ n | Theorem 4.8 (Nevanlinna-Levin) . If the function F ( z ) is holomorphic and of exponential type in thehalf-plane (cid:61) z ≥ , and if (4.6) holds, then1. F ( z ) ∞ (cid:89) k =1 − za k − za k = e iγ e u ( z )+ iv ( z ) , where u ( z ) = yπ (cid:90) ∞−∞ ln | F ( t ) | ( t − x ) + y dt + σ + F y,σ + F = h F ( π ) , v ( z ) is the harmonic conjugate of u ( z ) , and { a k } are the zeros of the function F ( z ) in the half-plane (cid:61) z > ;2. ln | F ( z ) | = yπ (cid:90) ∞−∞ ln | F ( t ) | ( t − x ) + y dt + σ + F y + ln | χ ( z ) | , z = x + iy, where χ ( z ) = ∞ (cid:89) k =1 − za k − za k . Proof. We refer the proof to [7, p. 240]. References [1] R. P. Boas, Entire Functions, Academic Press, New York, 1954.[2] S. Dyatlov and M. Zworski, Mathematical Theory of scattering resonances, V.1.0.http://math.mit.edu/ ∼ dyatlov/res/.[3] P. Deift and E. Trubowitz, Inverse scattering on the line, Commun. Pure Appl. Math., 32,121–251 (1979).[4] L. D. Faddeev, Properties of the S-matrix of the one-dimensional Schr¨odinger equation,Trudy Mat. Inst. Steklova, 73, 314–336 (1964). .[5] E. Korotyaev, Inverse resonance scattering for Schr¨odinger operator on the half line,Asymptot. Anal., 37, 215–226 (2004).[6] E. Korotyaev, Inverse resonance scattering on the real line, Inverse Probl., 21, 325–341(2005).[7] B. Ja. Levin, Distribution of Zeros of Entire Functions, revised edition, Translations ofMathematical Monographs, American Mathematical Society, 1972.78] B. Ja. Levin, Lectures on Entire Functions, Translation of Mathematical Monographs, V.150, AMS, Providence, 1996.[9] A. Melin, Operator methods for inverse scattering on the real line, Communications inPartial Differential Equations, Vol, 10, 677–766 (1985).[10] S. -H. Tang and M. Zworski, Potential Scattering on the Real Line, ”https://math.berkeley.edu/ ∼∼