Simplest examples of inverse scattering on the plane at fixed energy

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1 Simplest examples of inverse scattering on the plane at fixed energy Alexey Agaltsov, Roman Novikov To cite this version: Alexey Agaltsov, Roman Novikov. Simplest examples of inverse scattering on the plane at fixed energy. 17. <hal-1579> HAL Id: hal Submitted on 3 Jul 17 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Simplest examples of inverse scattering on the plane at fixed energy A. D. Agaltsov 1, R. G. Novikov July 3, 17 We consider the inverse scattering problem for the two-dimensional Schrödinger equation at fixed positive energy. Our results include inverse scattering reconstructions from the simplest scattering amplitudes. In particular, we give a complete analytic solution of the phased and phaseless inverse scattering problems for the single-point potentials (of the Bethe-Peierls-Fermi-Zeldovich-Berezin-Faddeev type). Then we study numerical inverse scattering reconstructions from the simplest scattering amplitudes using the Riemann-Hilbert-Manakov problem of the soliton theory. Finally, we apply the later numerical inverse scattering results for constructing related numerical solutions for equations of the Novikov-Veselov hierarchy at fixed positive energy. Keywords: inverse scattering, Schrödinger equation, numerical analysis, Novikov-Veselov equation Subjects: 35R3 (inverse problems for PDs), 65N1 (numerical analysis of inverse problems for PDs), 35P5 (scattering theory), 35J1 (Schrödinger operator), 35Q53 (KdV-like equations); 1 Introduction We consider the two-dimensional Schrödinger equation at fixed positive energy : ψ + v(x)ψ = ψ, x R, >, (1) where v is a real-valued sufficiently regular potential on R with sufficient decay at infinity. For this equation we consider the classical scattering solutions ψ + = ψ + (x, k), specified by the following asymptotics: ψ + (x, k) = e ikx + C( k ) ei k x x f(k, k x 1/ x ) + o( x 1 ), x +, x R, k R, k =, C( k ) = πi () πe iπ/ k 1, Dedicated to S. P. Novikov on the occasion of his 8th birthday 1 Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 3777 Göttingen, Germany; alexey.agaltsov@polytechnique.edu CMAP, cole Polytechnique, CNRS, Université Paris-Saclay, 9118, Palaiseau, France; IPT RAS, Moscow, Russia; novikov@cmap.polytechnique.fr 1

3 with a priori unknown coefficient f. The function f of () is known as the scattering amplitude for equation (1) and is defined on M = S 1 S 1, (3) S 1 = { m R m = }. () It is known that f possesses the following properties: f(k, l) = f( l, k) (reciprocity), (5) f(k, l) f(l, k) = π i f(k, m)f(l, m)dm (unitarity), (6) S 1 where k, l S 1. For possible assumptions on v assuring existence and uniqueness of ψ + at fixed k and properties (5), (6) for f see, e.g., [1, 6, 1]. Note also that f y (k, l) = e i(k l)y f(k, l), k, l S 1, (7) where f is the scattering amplitude for v and f y is the scattering amplitude for the translated potential v y = v( y), y R. For equation (1), the problem of finding ψ +, f from v is known as the direct scattering problem; the problem of finding v from f is known as the inverse scattering problem; and the problem of finding v from f is known as the phaseless inverse scattering problem. In addition to equation (1) we consider its isospectral deformations at fixed given by the Novikov-Veselov equation and its higher order analogs, see [16, 17, 6]. These equations admit a representation in the form of L-A-B Manakov triple (introduced in [11]), where L = + v. We recall that the first non-trivial equation of the Novikov-Veselov hierarchy can be written as: t v = Re ( zv 3 + z (vw) z w ), z w = 3 z v, w = w(x, t), w(x, t), x +, >, t R, x = (x 1, x ) R, (8) where v = v(x, t), w = w(x, t), z = 1 ( x 1 i x ), z = 1 ( x 1 + i x ). Analogs of the Gardner-Green-Kruskal-Miura relations for the equations of the Novikov-Veselov hierarchy and for the scattering amplitude f = f(k, l, t) are as follows: f(k, l, t) = exp ( it n+1 ( cos((n + 1)ϕk ) cos((n + 1)ϕ l ) )) f(k, l, ), k = (cos ϕ k, sin ϕ k ), l = (cos ϕ l, sin ϕ l ), (9) where n is the number of equation in the hierarchy; see [18] for the classical Gardner-Green-Kruskal-Miura relations and [13, 8] for (9).

4 Remark 1. (a) Properties (5), (6) are invariant with respect to transformations of f given by (7) and (9). (b) The differential scattering cross-section f is invariant with respect to transformations of f given by (7) and (9). In [13, 1] it was shown that if f is smooth, satisfies (5), (6) and f L (M ) < 3π, (1) then there is a smooth, real-valued, decaying at infinity potential v such that f is the scattering amplitude for v at fixed. In addition, this v is reconstructed from f via the algorithm suggested in [13, 1] and simplified in [15]. This final algorithm of [15] is recalled in Section. Note that these results of [13, 1, 15] are obtained using, in particular, the Riemann-Hilbert-Manakov problem of the soliton theory (see [1]) and results of [5] and [7]. In turn, the algorithm of [15] is implemented numerically in [3]. The results of the present work include inverse scattering reconstructions from some simplest functions f satisfying (5), (6) at fixed. In particular, in this framework we give a complete analytic solution of the phased and phaseless inverse scattering problems for the single-point potentials (of the Bethe-Peierls- Fermi-Zeldovich-Berezin-Faddeev type) v α,y (x), α R { }, y R, see Subsection.1. Then we give numerical inverse scattering reconstructions from some simplest scattering amplitudes f satisfying (5), (6) at fixed > using the numerical implementation (in MATLAB) of [3] of the algorithm recalled in Section. First of all, in this connection, we study reconstructions from constant f satisfying (5), (6) at fixed. In particular, such f arise as scattering amplitudes of the single-point potentials v α,y (x) for y = (i.e. supported at zero). Note that already for this simplest case there are no explicit analytic reconstruction formulas for regular potentials. Our numerical results for this case develop studies of [, ]. These results are presented in details in Subsection.. Then, using the numerical inverse scattering implementation of [3], we study reconstructions from functions f arising as scattering amplitudes of multi-point potentials (scatterers) v(x) = N v αj,y j (x), x R, α j R, y j R, (11) j=1 consisting of N single-point scatterers v αj,y j (x), where each point scatterer is described by its internal parameter α j and position y j (and y i y j for i j). This can be also considered as the first use of the multi-point potentials of the Bethe-Peierls-Fermi-Zeldovich-Berezin-Faddeev type for testing inverse scattering algorithms. Possibility of such tests was mentioned in [1]. These results are presented in details in Subsection 5.1. We emphasize that our aforementioned numerical reconstructions are obtained using the results of [15, 3] and can be considered as regular approximations to the initial multi-point potentials which are quite singular. 3

5 Finally, using the scattering amplitudes for the multi-point potentials, relations (9) and the inverse scattering implementation of [3] we obtain the related numerical solutions for equations of the Novikov-Veselov hierarchy. In particular, these results also illustrate non-uniqueness in the inverse scattering problem without phase information at fixed energy. See Subsection 5. for details. Inverse scattering algorithm It is convenient to use the following notations: z = x 1 + ix, z = x 1 ix, λ = 1/ (k 1 + ik ), λ = 1/ (l 1 + il ), (1) where x = (x 1, x ) R, k = (k 1, k ) S 1, l = (l 1, l ) S 1. notations k 1 = 1 1/ (λ + λ 1 ), k = i 1/ (λ 1 λ), where λ, λ T, l 1 = 1 1/ (λ + λ 1 ), l = i 1/ (λ 1 λ ), In these (13) T = { λ C λ = 1 }. (1) Using formulas (3), (), (1), (13), (1) one can see that S 1 = T, M = T T. (15) In addition, in these notations functions ψ +, f of () can be written as ψ + = ψ + (z, λ, ), f = f(λ, λ, ), (16) where λ, λ T, z C, >. The algorithm of [15] for finding v on R from f on M has the following scheme: and consists of the following steps: f h ± µ + µ v, (17) Step 1. Find functions h ± (λ, λ, ), λ, λ T, from the following linear integral equations: ( [ ]) h ± (λ, λ, ) π h ± (λ, λ λ, )χ ±i λ λ λ T (18) f(λ, λ, ) dλ = f(λ, λ, ), where χ(s) = { 1, s,, s <. (19)

6 Step. Solve the following linear integral equation for µ + (z, λ, ), z C, λ T, > : µ + (z, λ, ) + B(λ, λ, z, )µ + (z, λ, ) dλ = 1, () T where B(λ, λ, z, ) = 1 T 1 h + (ζ, λ, z, )χ T ( h (ζ, λ, z, )χ ( i [ ζ λ λ ζ [ ]) ζ i λ λ dζ ζ ζ λ(1 ) ]) dζ ζ λ(1 + ), (1) h ± (λ, λ, z, ) = h ± (λ, λ, ) ( ( exp (λ λ ) z (λ 1 λ 1 )z )), i () and λ, λ T, z C, >. Step 3. Define function µ (z, λ, ), z C, λ T, >, by the formula µ (z, λ, ) = µ + (z, λ, ) + πi h (λ, λ, z, ) T ( [ ]) λ χ i λ λ µ + (z, λ, ) dλ, λ where function h (λ, λ, z, ) is given by () and χ is defined by (19). Step. Potential v = v(x, ), x R, >, is given by the formula v(x, ) = z µ (z, ζ, ) dζ, () π where z = x 1 + ix, x = (x 1, x ), z = 1 ( x 1 i x ). T (3) Remark. As it was mentioned in the introduction, if f satisfies (5), (6), (1), then there exists a smooth real-valued decaying at infinity potential v such that f is the scattering amplitude for v at fixed >. In this result condition (1) can be replaced by a much weaker condition that all integral equations in (18), () are uniquely solvable. In addition, in notations of the present section condition (1) can be written as Note also that f L (T T ) < 1 3π. (5) v(x, τ ) = τ v(τx, ), >, τ >, (6) for v(x, ) reconstructed via (17) from f which is independent of, i.e. f = f(λ, λ ). 5

7 3 Scattering functions for multi-point potentials The scattering theory for multi-point potentials v mentioned in formula (11) of the introduction is presented, in particular, in [1, 9, 1]. In addition, all single-point potentials v α,y (x), α R \ {}, x, y R, can be considered as renormalizations of delta functions εδ(x y) with negative coefficients ε. We recall that for the multi-point potentials v of formula (11) the classical scattering functions ψ + and f are given by explicit formulas as follows. For the classical scattering eigenfunctions ψ + the following formulas hold: ψ + (x, k) = e ikx + N q + j (k)g+ (x y j, k), j=1 x R, k S 1, y j R, y j y m for j m, (7) G + (x, k) = i H(1) ( x k ), x R, k S 1, (8) where H (1) is the Hankel function of the first kind of order zero and q + (k) = ( q + 1 (k),..., q + N (k)) is the solution of the following linear system: A + (k)q + (k) = b + (k), (9) where A + (k) M N (C), b + (k) C N are given by { A + m,j (k) = 1 + αm π (πi ln k ), m = j, α m G + (y m y j, k), m j, (3) b + (k) = ( α 1 e iky1,..., α N e iky N ), α 1,..., α N R. (31) For the classical scattering amplitude f the following formula holds: f(k, l) = 1 (π) N q + j, k, l S 1 (k)e ilyj, (3) j=1 where q + j (k) are the same as in (7), (9). Reconstructions from constant f.1 Analytic inverse scattering for the single-point potentials The simplest functions on M are constants. Therefore, the results given below in this section are of particular interest. Lemma 1. Let f f on M for fixed >, where f is a complex constant. Then f satisfies (5), (6) if and only if f S, where S = { ζ C ζ ζ = iπ ζ ζ } = { ζ C ζ + i π = 1 π }. (33) 6

8 Lemma 1 follows from direct substitution of f into (5), (6). One can see that S is the circle centered at i 1 π of radius π. Using (9), (3) (for N = 1, y 1 = ) one can see that the scattering amplitude for the single point potential v α,y, y =, at fixed energy is given by the following formula: f(k, l) f α (), f α () = 1 α (π) 1 + α π (πi ln ), >, α R. (3) Assuming that f α () is defined as in (3), we have the following result. Theorem 1. Let ζ = f α () for fixed >. Then ζ S for any α R { }. Conversely, for any ζ S there exists the uniqie α R { } such that ζ = f α () and this α is given by the following formula: α = (π) ζ 1 πζ(πi ln ). (35) Proof of Theorem 1. The fact that the scattering amplitudes f (of Section 3) for the multi-point potentials satisfy (5), (6), the corollary of (3) that f = f α () is constant at fixed and α, and Lemma 1 imply that ζ S if ζ = f α (). The property that ζ S if ζ = f α () can also be verified by the direct calculation using the precise formula for f α () in (3). Conversely, consider the equation f α () = ζ with respect to α C { }, (36) for fixed ζ S and >. One can see that this equation is uniquely solvable and that the solution is given by formula (35). Direct calculations also show that α = ᾱ. Theorem 1 is proved. Remark 3. Theorem 1 gives a complete solution of the inverse scattering problem for the single point potentials v α,y, α R { }, y = (uniqueness, reconstruction, characterization). In view of formula (7), this solution admits a straightforward generalization to the case of v α,y with α R { }, y R. Remark. The property that ζ S if ζ = f α () can be considered as a relationship between the amplitude and phase of a single point scatterer v α,y, α R, y =. For a single point potential centered at zero, such a relationship was obtained in [, ] in the form: sin φ = β /, where β = β exp(iφ), β = (π) f(k, l). (37) However, in [, ] this relation is not yet related to the unitarity property (6) of the scattering amplitude f. 7

9 In addition, Theorem 1 implies the following corollary for inverse scattering without phase information. Corollary 1. Let > be fixed. Then for any σ [, 1 π ] the values of parameter α R { } such that σ = f α () are given by (35) with ζ = ± σ(1 σπ ) iσπ, (38) where the expression in (38) is single-valued for σ = and σ = 1 π and is double-valued for σ (, 1 π ). In addition, for any σ ( 1 π, ) there exists no α R { } such that σ = f α (). Remark 5. Let ζ = f α1 ( 1 ) for some fixed α 1 R { }, 1 >. Then for any > there exists the unique α R { } such that ζ = f α ( ), and this α is given by the following formula: α = α 1. (39) 1 + α1 π ln 1. Numerical reconstructions from f S In contrast to Theorem 1, we have no explicit formula for finding a regular potential v(x, ) with constant scattering amplitude f S \ {} at fixed energy >. However, the related numerical reconstructions using the numerical implementation of [3] of the algorithm recalled in Section are presented below in this section. It is convenient to use the following parametrization of the circle S of (33): Let Note that S = { ζ = ζ(ϕ) ζ(ϕ) = 1 π ( i + e iϕ ), ϕ [ π, π) }. () A ± = { ζ C ±(ζ + ζ + πζ ζ ln ) > } = { ζ C ± ( ζ + 1 π ln ) } (1) 1 π ln >. α < for ζ A S, α > for ζ A + S, 1 α = for ζ =, α = for ζ = π(πi ln ), () where α is given by (35) for fixed >. Figure 1 illustrates reconstructions v(x, ) from scattering amplitudes f(k, l) ζ(ϕ) for ϕ ( π, π ) as well as for ϕ ( π, 3π ), where ζ(ϕ) is given in () and = 1. We show the real parts of the reconstructed potentials v(x, ) only. The reason is that in our cases the imaginary parts are very small in comparison with the real parts. In addition, the domain of negative α for = 1 corresponds to ϕ (9, ), and α = corresponds to Note that numerical examples illustrating reconstructions from f(k, l) ζ(ϕ) with ϕ ( π, π ) at fixed > were already given in [] in the framework of inverse scattering in acoustics. 8

10 However, in addition to remarks of [], it is interesting to note that for the potential v(x, ) shown at Figure 1 for ϕ = 89 equation (1) can not be interpreted already as the acoustic Helmholtz equation with variable sound speed c(x) >. More precisely, in this case equation (1) can not be rewritten as where ψ ( ω c(x) v(x, ) = ( ω c(x) ω >, ω c The reason is that in this example ) ω c ψ = ω ψ, (3) c ) ω c, c(x) >, c >, =, = 1. () max v(x, ) >. (5) x In connection with the results of Subsection.1, it is important to note that the reconstruction shown in Figure 1 (right) is positive at zero, whereas all single-point potentials v α,y (x), α R { }, α, y R, can be considered as renormalized δ-functions εδ(x) with negative ε. On the other hand, the reconstruction shown in Figure 1 (left) looks indeed as a regularized εδ(x) with negative ε. To our knowledge, the reconstructions of Figure 1 (left) were not yet given in the literature. It is important to note that reconstructions shown in Figure 1 for ϕ = 1 and ϕ = 1 are obtained from scattering amplitudes f which differ only by their phases. However, these reconstructions differ by their signs as well as by the order of their amplitudes, and illustrate non-uniqueness in the phaseless inverse scattering problem (mentioned in the introduction) in the simplest case! Finally, note that condition (1) for f(k, l) ζ(ϕ) is fulfiled for ϕ (7.8, 19. ) only, where ζ is defined as in () and = 1. However, as it was already pointed out in [3, ], the algorithm recalled in Section works well much beyond limitation (1). 5 Further reconstruction examples Let f α1,...,α N,y 1,...,y N denote the scattering amplitude of (3) for fixed α 1,..., α N R { } and y 1,..., y N R. 5.1 Reconstructions from scattering amplitudes for multipoint potentials Developing results of Subsection. we also obtain numerical reconstructions v(x, ) from some scattering amplitudes f α1,...,α N,y 1,...,y N for multi-point potentials via the numerical implementation of [3] of the algorithm recalled above in Section. These reconstructions v(x, ) are illustrated by Figure for the case of 3-point potentials and = 1. In particular, in these examples we have 9

11 x x 1 Figure 1: Cross-sections of potentials v(x, ) (real part) numerically reconstructed from the scattering amplitudes f(k, l) ζ(ϕ), ϕ = 1 (α 5.15), 3 (α.98), (α.9) (left) and ϕ = 3 (α 1.5), 1 (α 1.86), 89 (α.71) (right). Here ζ(ϕ) is defined as in () and = 1. that the positions of points of the initial multi-point potentials are reconstructed very properly. In addition, we have v(x, ) N v αj,y j (x, ), (6) j=1 where v αj,y j (x, ) denote the reconstructions from the scattering amplitudes of the single-point potentials v αj,y j (x). 5. volutions according to equations of the Novikov-Veselov hierarchy We recall the following scheme, established in [7, 13, 1, 15], for constructing solutions v(x, t, ), x R, t R, of the Novikov-Veselov equation (8) and its higher order analogs at fixed > : consisting of the following steps: f(k, l) f(k, l, t) v(x, t, ), (7) Step 1. Given a smooth function f(k, l) on M satisfying (5), (6) and (1) at fixed >, define f(k, l, t), t R \ {}, using (9) with f(k, l, ) = f(k, l), where n is the number of equation in the Novikov-Veselov hierarchy. Step. Construct v = v(x, t, ) using scheme (17) of Section with f = f(k, l, t) for each fixed t R. Then v(x, t, ) satisfies the n-th equation of the Novikov-Veselov hierarchy at fixed >. 1

12 x x1 x1 x Figure : Numerical reconstructions from the scattering amplitudes of 3-point πk potentials fα1,α,α3,y1,y,y3 with yk = (3 cos( πk 3 ), 3 sin( 3 )). Left: α1 =.7, α = α3 = 1; right: α1 = α = α3 = 1. Here = 1. Note that in a similar way with Remark, condition (1) can be weakened to the condition of the unique solvability of all involved integral equations. In this section, using the numerical implementation of [3] of the algorithm recalled in Section, we present numerical solutions of the Novikov-Veselov equation (8) and its higher order analogs using scheme (7). In our examples we use some scattering amplitudes f of Section 3 for single- and multi-point potentials as the initial data of this scheme. Our numerical results are illustrated by Figures, 3,. In particular, Figure 3 shows v(x, t, ) for the case of single-point potential. In this case v(x,, ) looks as the reconstructions shown in Figure 1 (left) with min v(x,, ) x (8) Besides, Figures (right) and show v(x,, ) and v(x, t, ) for the case of 3-point potentials. Note that these numerical solutions v(x, t, ) of the Novikov-Veselov equation and its higher order analogs at fixed > illustrate the significant impact of the phase of the scattering amplitude f on the form of the reconstructed potential. These solutions (for fixed t) can be considered as non-trivial examples of non-uniqueness in the phaseless inverse scattering problem mentioned in the introduction. 6 Aknowledgements This work is partially supported by the PRC n 155 CNRS/RFBR: quations quasi-line aires, proble mes inverses et leurs applications. 11

13 x x x x 1 Figure 3: Solution v(x, t, ) of the n-th equation of the Novikov-Veselov hierarchy constructed from scattering amplitude f(k, l) f α (). Left: n = 1, α = 1, t = 3/ ; right: n =, α = 1, t = 5/. Here f α () is defined as in (3) and = 1. The color indicates the value varying from (black) to 1 (white) x x x x 1 Figure : Solution v(x, t, ) of the n-th equation of the Novikov-Veselov hierarchy constructed from scattering amplitude f(k, l) = f α1,α,α 3,y 1,y,y 3 (k, l), α 1 = α = α 3 = 1, y k = (3 cos( πk πk 3 ), 3 sin( 3 )). Left: n = 1, t =.5 3/ ; right: n = 5, t = 11/. Here = 1 and the color indicates the value varying from (black) to 1 (white). 1

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