Error estimates for phaseless inverse scattering in the Born approximation at high energies

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Error estimates for phaseless inverse scattering in the Born approximation at high energies

Alexey Agaltsov, Roman Novikov

To cite this version:

Alexey Agaltsov, Roman Novikov. Error estimates for phaseless inverse scattering in the Born approx- imation at high energies. The Journal of Geometric Analysis, Springer, 2020, 30 (3), pp.2340-2360.

�hal-01303885v2�

Error estimates for phaseless inverse scattering in the Born approximation at high energies ^∗

A. D. Agaltsov ¹ and R. G. Novikov ² December 8, 2016

Abstract. We study explicit formulas for phaseless inverse scatter- ing in the Born approximation at high energies for the Schr¨ odinger equation with compactly supported potential in dimension d ≥ 2.

We obtain error estimates for these formulas in the configuration space.

1 Introduction

We consider the time-independent Schr¨ odinger equation

− ∆ψ + v(x)ψ = Eψ, x ∈ R ^d , d ≥ 2, E > 0, (1) where

v ∈ L ^∞ (R ^d ), supp v ⊂ D, (2)

where D is some fixed open bounded domain in R ^d .

In quantum mechanics equation (1) describes an elementary particle inter- acting with a macroscopic object contained in D at fixed energy E. In this setting one usually assumes that v is real-valued.

Equation (1) at fixed E can be also interpreted as the Helmholtz equation of acoustics or electrodynamics. In these frameworks the coefficient v can be complex-valued. In addition, the imaginary part of v is related to the absorption coefficient.

For equation (1) we consider the classical scattering solutions ψ ⁺ = ψ ⁺ (x, k), where x = (x 1 , . . . , x d ) ∈ R ^d , k = (k 1 , . . . , k d ) ∈ R ^d , k ² = k ₁ ² + · · · + k _d ² = E.

These solutions ψ ⁺ can be specified by the following asymptotics as |x| → ∞:

ψ ⁺ (x, k) = e ^ikx + c(d, |k|) e ^{i | k || x |}

|x| ^(d−1)/2 f (k, |k| _|x| ^x ) + O(|x| ^{− (d+1)/2} ), x ∈ R ^d , k ∈ R ^d , k ² = E, kx = k 1 x 1 + · · · + k d x d ,

c(d, |k|) = −πi(−2πi) ^{(d − 1)/2} |k| ^{(d − 3)/2} ,

(3)

for some a priori unknown f . The function f arising in (3) is defined on M _E =

(k, l) ∈ R ^d × R ^d : k ² = l ² = E , (4)

∗

Dedicated to G. M. Henkin

1

CMAP, Ecole Polytechnique, CNRS, Universit´ e Paris-Saclay, 91128, Palaiseau, France;

email: agaltsov@cmap.polytechnique.fr

2

CMAP, Ecole Polytechnique, CNRS, Universit´ e Paris-Saclay, 91128, Palaiseau, France;

IEPT RAS, 117997 Moscow, Russia; email: novikov@cmap.polytechnique.fr

and is known as the classical scattering amplitude for equation (1).

In quantum mechanics |f (k, l)| ² describes the probability density of scatter- ing of particle with initial momentum k into direction l/|l| 6= k/|k|, and is known as differential scattering cross section for equation (1); see, e.g., [11, Chapter 1, Section 6].

The problem of finding ψ ⁺ and f from v is known as the direct scattering problem for equation (1). For solving this problem, one can use, in particu- lar, the Lippmann-Schwinger integral equation for ψ ⁺ and an explicit integral formula for f , see, e.g., [5, 10, 29].

In turn, the problem of finding v from f is known as the inverse scattering problem (with phase information) and the problem of finding v from |f| ² is known as the phaseless inverse scattering problem for equation (1).

There are many important results on the former inverse scattering problem with phase information; see [3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 22, 23, 24, 25, 28] and references therein. In particular, it is well known that the scattering amplitude f uniquely determines v via the Born approximation formulas at high energies:

b v(k − l) = f(k, l) + O(E ⁻

), E → +∞, (k, l) ∈ M E , (5) v(p) = (2π) b ^−d

Z

R^d

e ^ipx v(x) dx, p ∈ R ^d , (6) and the inverse Fourier transform; see, e.g., [9, 28].

On the other hand, the literature for the phaseless case is much more limited;

see [7, 29] and references therein for the case of the aforementioned phaseless inverse problem and see [19, 20, 21, 26, 27, 29] and references therein for the case of some similar inverse problems without phase information. In addition, it is well known that the phaseless scattering data |f | ² does not determine v uniquely, even if |f| ² is given completely for all positive energies. In particular, it is known that

f y (k, l) = e ^{i(k − l)y} f (k, l),

|f y (k, l)| ² = |f (k, l)| ² , k, l ∈ R ^d , k ² = l ² > 0, (7) where f is the scattering amplitude for v and f y is the scattering amplitude for v y = v(· − y), where y ∈ R ^d ; see [29] and references therein.

In the present work, in view of the aforementioned non-uniqueness for the problem of finding v from |f | ² , we consider the modified phaseless inverse scat- tering problem formulated below as Problem 1. Let

S = {|f | ² , |f 1 | ² , . . . , |f m | ² }, (8) where f is the scattering amplitude for v and f 1 , . . . , f m are the scattering amplitudes for v 1 , . . . , v m , where

v j = v + w j , j = 1, . . . , m, (9)

where w 1 , . . . , w m are additional a priori known background scatterers such that

w j ∈ L ^∞ (R ^d ), supp w j ⊂ Ω j ,

Ω j is an open bounded domain in R ^d , Ω j ∩ D = ∅, w j 6= 0, w j

6= w j

if j 1 6= j 2 (in L ^∞ ( R ^d )),

(10)

where j, j 1 , j 2 ∈ {1, . . . , m}. Thus, S consists of the phaseless scattering data

|f | ² , |f 1 | ² , . . . , |f m | ² measured sequentially, first, for the unknown scatterer v and then for v in the presence of known scatterer w j disjoint from v for j = 1, . . . , m.

Actually, in the present work we continue studies of [29] on the following inverse scattering problem for equation (1):

Problem 1. Reconstruct potential v from the phaseless scattering data S for some appropriate background scatterers w 1 , . . . , w m .

Studies of Problem 1 in dimension d ≥ 2 were started in [29]. In dimension d = 1 for m = 1 studies of Problem 1 were started earlier in [2], where phaseless scattering data was considered for all E > 0.

Actually, the key result of [29] consists in a proper extension of formula (5) for the Fourier transform b v of v to the phaseless case of Problem 1; see Section 2.

In the present work we proceed from the aforementioned result of [29] and study related approximate reconstruction of v in the configuration space. In this connection our results consist in obtaining related error estimates in the configuration space at high energies E; see Section 4.

In addition, results of the present work are necessary for extending the iter- ative algorithm of [28] to the phaseless case of Problem 1. The latter extension will be given in [1].

2 Extension of formula (5) to the phaseless case

Actually, the key result of [29] consists in the following formulas for solving Problem 1 in dimension d ≥ 2 for m = 2 at high energies E:

Re b v Im b v

= 1 2

Re w b 1 Im w b 1

Re w b 2 Im w b 2

− 1

| b v 1 | ² − | b v| ² − | w b 1 | ²

| b v 2 | ² − | b v| ² − | w b 2 | ²

, (11)

| b v j (p)| ² = |f j (k, l)| ² + O(E ⁻

), E → +∞,

p ∈ R ^d , (k, l) ∈ M E , k − l = p, j = 0, 1, 2, (12) where:

• v 0 = v, v j is defined by (9), j = 1, 2, and f 0 = f , f 1 , f 2 are the scattering amplitudes for v 0 , v 1 , v 2 , respectively;

• b v = b v(p), b v j = b v j (p), w b j = w b j (p), p ∈ R ^d , are the Fourier transforms of v,

v j , w j (defined as in (6));

• formula (11) is considered for all p ∈ R ^d such that the determinant ζ w

, w

(p) == Re ^def w b 1 (p) Im w b 2 (p) − Im w b 1 (p) Re w b 2 (p) 6= 0. (13) The point is that using formulas (12) for d ≥ 2 with

k = k E (p) = ^p ₂ + E − ^p ₄

1/2

γ(p), l = l E (p) = − ^p ₂ + E − ^p ₄

1/2

γ(p),

|γ(p)| = 1, γ(p)p = 0,

(14)

where p ∈ R ^d , |p| ≤ 2 √

E, one can reconstruct | b v| ² , | b v 1 | ² , | b v 2 | ² from S at high energies for any p ∈ R ^d . And then using formula (11) one can reconstruct v b completely, provided that condition (13) is fulfilled for almost all p ∈ R ^d . Remark 1. Formulas (12) can be precised as formula (2.15) of [29]:

| b v j (p)| ² − |f j (k, l)| ²

≤ c(D j )N _j ³ E ⁻

,

p = k − l, (k, l) ∈ M _E , E

≥ ρ(D j , N j ), j = 0, 1, 2, (15) where kv j k _L

(D

) ≤ N j , j = 0, 1, 2, and D 0 = D, D j = D ∪ Ω j , j = 1, 2, and constants c, ρ are given by formulas (3.10) and (3.11) in [29] (and, in particular, ρ ≥ 1).

In addition, from the experimental point of view it seems to be, in particular, convenient to consider Problem 1 with m = 2 for the case when w 2 is just a translation of w 1 :

w 2 (x) = w 1 (x − y), x ∈ R ^d , y ∈ R ^d . (16) In this case

w b 2 (p) = e ^ipy w b 1 (p), ζ w

, w

(p) = sin(py)| w b 1 (p)| ² , p ∈ R ^d . (17) On the level of analysis, the principal complication of (11), (12) in compar- ison with (5) consists in possible zeros of the determinant ζ w

, w

of (13). For some simplest cases, we study these zeros in the next section.

3 Zeros of the determinant ζ

w b 1 , w b 2

Let

Z w

, w

=

p ∈ R ^d : ζ w

, w

(p) = 0 , Z w

=

p ∈ R ^d : w b j (p) = 0 , j = 1, 2, (18) where ζ is defined by (13). From (13), (18) it follows that

Z w

∪ Z w

⊆ Z w

, w

. (19)

In view of (19), in order to construct examples of w 1 , w 2 such that the set

Z w

, w

is as simple as possible, we use the following lemma:

Lemma 1. Let

w(x) = |x| ^ν K ν (|x|) Z

R^d

q(x − y)q(y) dy, x ∈ R ^d , ν > 0, (20) K ν (s) = Γ( ¹ ₂ + ν)

√ π 2

s ^ν Z ∞

0

cos(st) dt

(1 + t ² )

^+ν , s > 0, (21) q ∈ L ^∞ (R ^d ), q = q, q 6= 0 in L ^∞ (R ^d ),

q(x) = 0 if |x| > r, q(x) = q(−x), x ∈ R ^d . (22) Then

w ∈ C( R ^d ), w = w, w(x) = 0 if |x| > 2r, x ∈ R ^d ,

w(p) = b w(p) b ≥ c 1 (1 + |p|) ^{− β} , p ∈ R ^d , (23) for β = d + 2ν and some positive constant c 1 = c 1 (q, ν ), where w b is the Fourier transform of w. In addition, if q ≥ 0, then w ≥ 0.

We recall that K ν defined by (21) is the modified Bessel function of the second kind and order ν. In addition, Γ denotes the gamma function.

Lemma 1 is proved in Section 8.

As a corollary of Lemma 1, functions

w j (x) = w(x − T j ), x ∈ R ^d , T j ∈ R ^d , (24) where w is constructed in Lemma 1, give us examples of w j satisfying (10) for fixed D, Ω j and for appropriate radius r of Lemma 1 and translations T j of (24), and such that

Z w

= ∅,

| w b j (p)| = w(p) b ≥ c 1 (1 + |p|) ^{− β} , p ∈ R ^d , (25) where c 1 , β are the same as in (23). In addition,

ζ w

, w

(p) = sin(py)| w(p)| b ² , y = T 2 − T 1 6= 0, p ∈ R ^d , Z w

, w

=

p ∈ R ^d : sin(py) = 0 =

p ∈ R ^d : py ∈ π Z , (26) for w 1 , w 2 of (24).

As another corollary of Lemma 1, we have that

if w 1 is defined as in (24) and w 2 = iw 1 , then ζ w

, w

(p) = | w(p)| b ² ≥ c ² ₁ (1 + |p|) ^{− 2β} , p ∈ R ^d ,

Z w

, w

= ∅.

(27)

We recall that complex-valued v and w j naturally arise if we interpret equa- tion (1) for fixed E as the Helmholtz equation of acoustics or electrodynamics.

Finally, note that

Z w

,..., w

= ^π _s Z ^d , where

Z w

,..., w

= Z w

, w

∩ Z w

, w

∩ · · · ∩ Z w

, w

, (28)

if w 1 is defined as in (24), and

w 2 (x) = w 1 (x − se 1 ), . . . , w d+1 (x) = w 1 (x − se d ), (29) where (e 1 , . . . , e d ) is the standard basis of R ^d and s > 0.

Thus, in principle, for Problem 1 with background scatterers w 1 , . . . , w d+1

as in (29), for each p ∈ R ^d \ ^π _s Z ^d formulas (11), (12) can be used with appropriate w j in place of w 2 , where j = 2, . . . , d + 1.

4 Error estimates in the configuration space

We recall that for inverse scattering with phase information the scattering am- plitude f on M E processed by (5) and the inverse Fourier transform yield the approximate reconstruction

u(·, E) = v + O(E ^−α ) in L ^∞ (D) as E → +∞, α = n − d

2n , (30)

if v ∈ W ^n,1 (R ^d ), n > d (in addition to the initial assumption (2)), where W ^n,1 (R ^d ) denotes the standard Sobolev space of n-times differentiable functions in L ¹ (R ^d ):

W ^n,1 ( R ^d ) =

u ∈ L ¹ ( R ^d ) : kuk n,1 < ∞ , kuk n,1 = max

|J|≤n

∂ ^|J| u

∂x ^J _L

₍

R^d

)

, n ∈ N ∪ {0}. (31) More precisely, the approximation u(·, E) in (30) is defined by

u(x, E) = Z

B

e ^{− ipx} f (k E (p), l E (p)) dp, x ∈ D, r(E) = 2τ E

for some fixed τ ∈ (0, 1],

(32)

where

B _r =

p ∈ R ^d : |p| ≤ r , (33)

α is defined in (30), and k E (p), l E (p) are defined as in (14) with some piecewise continuous vector-function γ on R ^d ; see, e.g., [28]. In addition, estimate (30) can be precised as

|u(x, E) − v(x)| ≤ A(D, N, M, d, n, τ )E ^{− α} , x ∈ D, E

≥ ρ(D, N ), (34) where kvk L

(D) ≤ N , kvk n,1 ≤ M , ρ is the same as in (15) and the expression for A can be found in formula (3.10) of [28].

Analogs of u(·, E) for the phaseless case are given below in this section. In

particular, related formulas depend on the zeros of determinant ζ w

, w

of (13).

We consider

U w

, w

= Re U w

, w

+ i Im U w

, w

, Re U w

, w

(p, E)

Im U w

, w

(p, E)

= ¹ ₂ M ^{− 1}

w

, w

(p)b w

, w

(p, E), (35) M w

, w

(p) =

Re w b 1 (p) Im w b 1 (p) Re w b 2 (p) Im w b 2 (p)

, (36)

M ⁻¹

w

, w

(p) = 1 ζ w

, w

(p)

Im w b 2 (p) − Im w b 1 (p)

− Re w b 2 (p) Re w b 1 (p)

, (37)

b w

, w

(p, E) =

|f 1 (p, E)| ² − |f (p, E)| ² − | w b 1 (p)| ²

|f 2 (p, E)| ² − |f (p, E)| ² − | w b 2 (p)| ²

, (38)

f (p, E) = f (k E (p), l E (p)), f j (p, E) = f j (k E (p), l E (p)), j = 1, 2, (39) where w b 1 , w b 2 , f , f 1 , f 2 are the same as in (11), (12), ζ w

, w

is defined by (13), k E (p), l E (p) are the same as in (14), (32), and p ∈ B ₂ ^√ _E , d ≥ 2.

For Problem 1 for d ≥ 2, m = 2, and for the case when ζ

w

, w

has no zeros (the case of (27) in Section 3) we have the following result:

Theorem 1. Let v satisfy (2) and v ∈ W ^n,1 ( R ^d ) for some n > d. Let w 1 , w 2

be the same as in (27). Let u(x, E) =

Z

B

e ^{− ipx} U w

, w

(p, E) dp, x ∈ D,

r 1 (E) = 2τ E

, α 1 = n − d

2(n + β ) , for some fixed τ ∈ (0, 1],

(40)

where U w

, w

is defined by (35), B _r is defined by (33), β is the number of (23), (27). Then

u(·, E) = v + O(E ^{− α}

) in L ^∞ (D), E → +∞,

|u(x, E) − v(x)| ≤ A 1 E ^{− α}

, x ∈ D, E

≥ ρ 1 , (41) where ρ 1 and A 1 are defined in formulas (55) and (64) of Section 5.

Theorem 1 is proved in Section 5.

Next, we set Z ^ε

w

, w

=

p ∈ R ^d : py ∈ (−ε, ε) + πZ , y ∈ R ^d \ 0, 0 < ε < 1, (42) where w b 1 , w b 2 and y are the same as in (24)–(26). One can see that Z ^ε

w

, w

is the open _| ^ε _{y |} -neighborhood of Z w

, w

defined in (26).

Note that

for any p ∈ Z ^ε

w

, w

there exists

the unique z(p) ∈ Z such that |py − πz(p)| < ε. (43)

(x ₂ , . . . , x _d )

x ₁

π

| y | . . . ^πz(p) | y |

p

p ^ε ₋ p ^ε ₊

p _⊥

y O

ε

| y | ε

| y |

Figure 1: Vectors p, p _⊥ , y and p ^ε _± of formula (44)

In addition to U w

, w

of (35), we define U ^ε

w

, w

(p, E) = ¹ ₂ U w

, w

(p ^ε ₋ , E) + U w

, w

(p ^ε ₊ , E) , p ^ε _± = p _⊥ + πz(p) _|y| ^y

± ε _|y| ^y

, p _⊥ = p − (py) _|y| ^y

, p ∈ B ₂ ^√ _E ∩ Z ^ε

w

, w

, (44) where z(p) is the integer number of (43). The geometry of vectors p, p _⊥ , y, p ^ε _± is illustrated in Fig. 1 for the case when the direction of y coincides with the basis vector e 1 = (1, 0, . . . , 0).

For Problem 1 for d ≥ 2, m = 2, and for the case when ζ w

, w

has zeros on hyperplanes (the case of (26) in Section 3) we have the following result:

Theorem 2. Let v satisfy (2) and v ∈ W ^n,1 (R ^d ) for some n > d. Let w 1 , w 2

be the same as in (24)–(26). Let

u(x, E) = u 1 (x, E) + u 2 (x, E), x ∈ D, u 1 (x, E) =

Z

B

\Z

wb1,wb2

e ^−ipx U

w

, w

(p, E) dp, u 2 (x, E) =

Z

B

∩ Z

wb1,wb2

e ^{− ipx} U ^ε

w

, w

(p, E) dp, r 2 (E) = 2τ E

, ε 2 (E) = E ^{− α 2}

, α 2 = ^{n − d}

2 n+β+ n − d 2

, for some fixed τ ∈ (0, 1],

(45)

where U w

, w

and U ^ε

w

, w

are defined by (35) , (44) , B r and Z ^ε

w

, w

are defined

by (33), (42), and β is the number of (23). Then

u(·, E) = v + O(E ^{− α}

) in L ^∞ (D), E → +∞,

|u(x, E) − v(x)| ≤ A 2 E ^{− α}

, x ∈ D, E

≥ ρ 2 , (46) where ρ 2 and A 2 are defined in formulas (65) and (85) of Section 6.

Theorem 2 is proved in Section 6.

Next, we set Z ^ε

w

,..., w

= B _ε/s ⁰ + ^π _s Z ^d , 0 < ε < 1,

B _r ⁰ = B _r \ ∂B r , r > 0, (47) where w b 1 , . . . , w b d+1 are the same as in (28), (29), and B _r is defined by (33).

One can see that Z ^ε

w

,..., w

is the open ^ε _s -neighborhood of Z w

,..., w

defined in (28).

Note that

for any p ∈ Z ^ε

w

,..., w

there exists

the unique z(p) ∈ Z ^d such that |sp − πz(p)| < ε. (48) In addition, we consider i ⁰ such that

i ⁰ = i ⁰ (p, s), p = (p 1 , . . . , p d ) ∈ R ^d \ ^π _s Z ^d , s > 0, i ⁰ take values in {2, . . . , d + 1},

| sin(sp i

− 1 )| ≥ | sin(sp i − 1 )| for all i ∈ {2, . . . , d + 1}.

(49)

Let

U w

,..., w

(p, E) = U

w

, w

(p, E), p ∈ R ^d \ ^π _s Z ^d , (50) U ^ε

w

,..., w

(p, E) = 1

| S ^d−1 | Z

S^d−1

U w

,..., w

( ^ε _s ϑ + ^π _s z(p), E) dϑ, p ∈ Z ^ε

w

,..., w

, (51) S ^{d − 1} =

p ∈ R ^d : |p| = 1 , (52)

where | S ^{d − 1} | denotes the standard Euclidean volume of S ^{d − 1} , U w

, w

is defined in a similar way with U w

, w

of (35), w b 1 , . . . , w b d+1 are the same as in (28), (29), and i ⁰ is the same as in (49).

For Problem 1 for d ≥ 2, m = d + 1, we also have the following result:

Theorem 3. Let v satisfy (2) and v ∈ W ^n,1 (R ^d ) for some n > d. Let w 1 , . . . ,

w d+1 be the same as in (29). Let

u(x, E) = u 1 (x, E) + u 2 (x, E), x ∈ D, u 1 (x, E ) =

Z

B

\Z

wb1,...,wd+1b

e ^−ipx U w

,..., w

(p, E) dp,

u 2 (x, E ) =

Z

B

∩ Z

wb1,...,wd+1b

e ^{− ipx} U ^ε

w

,..., w

(p, E) dp, r 3 (E) = 2τ E

, ε 2 (E) = E ⁻

,

α 3 = ^{n − d}

2 n+β+ n − d d+1

, for some fixed τ ∈ (0, 1],

(53)

where U w

,..., w

and U ^ε

w

,..., w

are defined by (50), (51), B r and Z ^ε

w

,..., w

are defined by (33), (47), and β is the number of (23). Then u(·, E) = v + O(E ^{− α}

) in L ^∞ (D), E → +∞,

|u(x, E) − v(x)| ≤ A 3 E ^{− α}

, x ∈ D, E

≥ ρ 3 , (54) where ρ 3 and A 3 are defined in formulas (86) and (99) of Section 7.

Theorem 3 is proved in Section 7.

5 Proof of Theorem 1

Proposition 1. Let v satisfy (2) and w 1 , w 2 be the same as in (27), d ≥ 2.

Then:

b v(p) − U

w

, w

(p, E)

≤ c 2 | w(p)| b ⁻¹ E ⁻

for p ∈ B ₂ ^√ _E , E

≥ ρ 1 , c 2 = 2c(D 0 )N ₀ ³ + 2c(D 1 )N ₁ ³ ,

ρ 1 = max

j=0,1 ρ(D j , N j ),

(55)

where w is the function of (23), (24), and c, ρ, N j , D j , j = 0, 1, 2, are the same as in estimates (15).

Proposition 1 follows from formulas (11), (14), estimates (15), definitions (35)–(39), and the properties that

Ω 2 = Ω 1 , D 2 = D 1 , N 2 = N 1 . (56)

In turn, properties (56) follow from (9) for j = 1, 2, (10) for j = 1, and from

the equality w 2 = iw 1 assumed in (27).

Next, we represent v as follows:

v(x) = v ⁺ (x, r) + v ⁻ (x, r), x ∈ D, r > 0, v ⁺ (x, r) =

Z

B

e ^{− ipx} b v(p) dp, v ⁻ (x, r) =

Z

R^d

\B

e ^−ipx b v(p) dp.

(57)

Since v ∈ W ^n,1 (R ^d ), n > d, we have

|v ⁻ (x, r)| ≤ c 3 kvk n,1 r ^{d − n} , x ∈ D, r > 0,

c 3 = | S ^{d − 1} | ^(2π) _{n −}

_d ^d

, (58) where k · k _n,1 is defined in (31), and | S ^{d − 1} | is the standard Euclidean volume of S ^{d − 1} . Indeed,

|p ^k ₁

· · · p ^k _d

v(p)| ≤ b (2π) ^{− d} kvk n,1 , p = (p 1 , . . . , p d ) ∈ R ^d ,

for any k 1 , . . . k d ∈ N ∪ {0}, k 1 + · · · + k d ≤ n, (59) assuming also that p ⁰ _j = 1. Taking an appropriate sum in (59) over all such k 1 , . . . , k d with k 1 + · · · + k d = m ≤ n, we get

|p| ^m | b v(p)| ≤ (|p 1 | + · · · + |p d |) ^m | v(p)| ≤ b (2π) ^{− d} d ^m kvk n,1 , p ∈ R ^d . (60) The definition of v ⁻ of (57) and inequalities (60) for m = n imply (58).

In addition, using Proposition 1 and the estimate on w b of (23), we obtain:

v ⁺ (x, r) − Z

B

e ^{− ipx} U w

, w

(p, E) dp

≤ c 2 E ⁻

Z

B

| w(p)| b ^{− 1} dp

≤ c ⁻ ₁ ¹ c 2 E ⁻

Z

B

(1 + |p|) ^β dp ≤ c 4 E ⁻

r ^d+β , c 4 = | S ^d−1 | ² _d+β

c ⁻ ₁ ¹ c 2 , x ∈ D, 1 ≤ r ≤ 2E

, E

≥ ρ 1 .

(61)

As a corollary of (57), (58), (61), we have

v(x) − Z

B

e ^−ipx U w

, w

(p, E) dp

≤ c 3 kvk n,1 r ^d−n + c 4 E ⁻

r ^d+β , (62) where x ∈ D, 1 ≤ r ≤ 2E

, E

≥ ρ 1 . In addition, if r = r 1 (E), where r 1 (E) is defined in (40), then

r ^d−n = (2τ) ^d−n E ^−α

,

E ⁻

r ^d+β = (2τ) ^d+β E ^{− α}

. (63) Using formulas (62) and (63) and taking into account definitions (40), we obtain

|u(x, E) − v(x)| ≤ A 1 E ^{− α}

, x ∈ D, E

≥ ρ 1

A 1 = A 1 (D 0 , D 1 , N 0 , N 1 , M, d, n, β, τ )

= (2τ ) ^{d − n} c 3 kvk n,1 + (2τ) ^d+β c 4 ,

(64)

where D j , N j , j = 0, 1, are the same as in estimates (15) and kvk n,1 ≤ M . Theorem 1 is proved.

6 Proof of Theorem 2

Proposition 2. Let v satisfy (2) and w 1 , w 2 be the same as in (24)–(26), d ≥ 2. Then:

v(p) b − U w

, w

(p, E)

≤ c 5 ε ^{− 1} (1 + |p|) ^β E ⁻

, p ∈ B ₂ ^√ _E \ Z ^ε

w

, w

, E

≥ ρ 2 , 0 < ε < 1, c 5 = ^π ₂ (2c(D 0 )N ₀ ³ + c(D 1 )N ₁ ³ + c(D 2 )N ₂ ³ )c 1 ,

ρ 2 = max

j=0,1,2 ρ(D j , N j ),

(65)

in addition, if v ∈ W ^n,1 ( R ^d ), n ≥ 0, then:

| b v(p) − U ^ε

w

, w

(p, E)|

≤ 2 ^β c 5 ε ^{− 1} (1 + |p|) ^β E ⁻

+ c 6 ε(1 + _| ^π _{y |} |z(p)| + |p _⊥ |) ^{− n} , p ∈ B ₂ ^√ _E ∩ Z ^ε

w

, w

, E ≥ ρ 2 , 0 < ε < min{1, ¹ ₂ |y|};

(66)

c 6 = 2 ^{n (d+1)} _(2π)

| y | max

j=1,...,d kx j vk n,1 ,

where c, ρ, N j , D j , j = 0, 1, 2, are the same as in estimates (15), c 1 , β are the same as in Lemma 1; z(p), p _⊥ are defined in (43), (44), x j v = x j v(x), and k · k _n,1 is defined in (31).

Proof of Proposition 2. It follows from formulas (24), (26) and (36), (37) that M w

, w

(p) = w(p) b

cos(T 1 p) sin(T 1 p) cos(T 2 p) sin(T 2 p)

, p ∈ R ^d , M ⁻¹

w

, w

(p) = 1 sin(py) w(p) b

sin(T 2 p) − sin(T 1 p)

− cos(T 2 p) cos(T 1 p)

, p ∈ R ^d \ Z ^ε

w

, w

. (67)

Also note that

| sin(py)| ≥ ^2ε _π , p ∈ R ^d \ Z ^ε

w

, w

, 0 < ε < 1. (68) The estimate (65) follows from (11), (15), (23), (35), (38), (39) and from (67), (68).

It remains to prove (66). Using definition (44), one can write U ^ε

w

, w

(p, E) − b v(p) = ϕ ^ε ₁ (p, E) + ϕ ^ε ₂ (p), p ∈ B ₂ ^√ _E ∩ Z ^ε

w

, w

, ϕ ^ε ₁ (p, E) = ¹ ₂ U ^ε

w

, w

(p ^ε ₋ , E) − b v(p ^ε ₋ ) + ¹ ₂ U ^ε

w

, w

(p ^ε ₊ , E) − b v(p ^ε ₊ )

, p ∈ B ₂ ^√ _E ∩ Z ^ε

w

, w

, ϕ ^ε ₂ (p) = ¹ ₂ b v(p ^ε ₋ ) + v(p b ^ε ₊ )

− b v(p), p ∈ Z ^ε

w

, w

.

(69)

Using estimate (65), formula (69) and the definitions of p ^ε _± in (44), we get

|ϕ ^ε ₁ (p, E)| ≤ ¹ ₂ c 5 ε ⁻¹ E ⁻

(1 + |p ^ε ₋ |) ^β + (1 + |p ^ε ₊ |) ^β

≤ c 5 ε ^{− 1} (1 + |p| + 2 _| ^ε _{y |} ) ^β E ⁻

≤ 2 ^β c 5 ε ^{− 1} (1 + |p|) ^β E ⁻

, for ε as in (66), p ∈ B ₂ ^√ _E ∩ Z ^ε

w

, w

.

(70)

Next, using the definition of ϕ ^ε ₂ in (69) and the mean value theorem, we obtain

|ϕ ^ε ₂ (p)| ≤ _| ^ε _{y |} max{| _| ^y _{y |} ∇ b v(ξ)|: ξ ∈ [p ^ε ₋ , p ^ε ₊ ]}, p ∈ Z ^ε

w

, w

, (71) where [p ^ε ₋ , p ^ε ₊ ] denotes the segment joining p ^ε ₋ to p ^ε ₊ . Here, the mean value theorem was used for b v(ξ) on [p ^ε ₋ , p] and on [p, p ^ε ₊ ].

Note also that

|∇ b v(ξ)| ≤ d max

j=1,...,d

_∂ξ ^∂

^v

j

(ξ)

, ξ = (ξ 1 , . . . , ξ d ) ∈ [p ^ε ₋ , p ^ε ₊ ]. (72) In addition, the following estimates hold:

∂ b v

∂ξ j (ξ)

≤ (1 + d) ⁿ

(2π) ^d (1 + |ξ|) ⁿ kx j vk n,1 , ξ ∈ [p ^ε ₋ , p ^ε ₊ ], j = 1, . . . , d. (73) Indeed, taking the sum in (60) over all m = 0, . . . , n with the binomial coeffi- cients, we get

(1 + |p|) ⁿ | b v(p)| ≤ (1 + |p 1 | + · · · + |p d |) ⁿ | b v(p)|

≤ (2π) ^−d (1 + d) ⁿ kvk n,1 , p ∈ R ^d . (74) Estimates (73) follow from (74), where we replace v by x j v and use that v belongs to W ^n,1 ( R ^d ) and is compactly supported.

Estimates (71)–(73) imply

|ϕ ^ε ₂ (p)| ≤ 2 ^{− n} c 6 ε max

1 + |ξ| − n

: ξ ∈ [p ^ε ₋ , p ^ε ₊ ] , p ∈ Z ^ε

w

, w

. (75) Using also that

ξ = τ _| ^y _{y |} + p _⊥ , where |τ − _| ^π _{y |} z(p)| ≤ _| ^ε _{y |} , if ξ ∈ [p ^ε ₋ , p ^ε ₊ ], (76) and that ε < |y|, we obtain

|ϕ ^ε ₂ (p)| ≤ 2 ^{− n} c 6 ε 1 + ¹ ₂ ( _| ^π _{y |} |z(p)| − _| ^ε _{y |} + |p _⊥ |) − n

≤ c 6 1 + _|y| ^π |z(p)| + |p _⊥ | − n

, p ∈ Z ^ε

w

, w

. (77) Estimate (66) follows from (70) and (77).

Proposition 2 is proved.

The final part of the proof of Theorem 2 is as follows. In a similar way with (57), we represent v as follows:

v(x) = v ⁺ ₁ (x, r) + v ⁺ ₂ (x, r) + v ⁻ (x, r), x ∈ D, r > 0, v ₁ ⁺ (x, r) =

Z

B

\Z

wb1,wb2

e ^{− ipx} v(p) b dp, v ₂ ⁺ (x, r) =

Z

B

∩Z

wb1,wb2

e ^{− ipx} v(p) b dp, v ⁻ (x, r) =

Z

R^d

\B

e ^{− ipx} b v(p) dp.

(78)

Since v ∈ W ^n,1 (R ^d ), estimate (58) holds.

Using estimates (65), (66), we get:

v ⁺ ₁ (x, r) − Z

B

\ Z

wb1,wb2

e ^{− ipx} U w

, w

(p, E) dp +v ₂ ⁺ (x, r) −

Z

B

∩ Z

wb1,wb2

e ^{− ipx} U ^ε

w

, w

(p, E) dp

≤ I 1 + I 2 ,

(79)

I 1 = 2 ^β c 5 ε ^{− 1} E ⁻

Z

B

(1 + |p|) ^β dp, (80)

I 2 = c 6 ε Z

B

∩Z

wb1,wb2

1 + _| ^π _{y |} |z(p)| + |p _⊥ | − n

dp, (81)

x ∈ D, 1 ≤ r ≤ 2E

, E

≥ ρ 2 , where ρ 2 is the same as in Proposition 2. In addition:

I 1 ≤ c 7 ε ^{− 1} E

r ^d+β , c 7 = | S ^{d − 1} | ² _d+β

c 5 ; (82) I 2 = c 6 ε X

z ∈Z

Z

(

τ

+p

≤r

| τ − π

|y| z |≤ ε

|y|

)

1 + _| ^π _{y |} |z| + |p _⊥ | − n

dτ dp _⊥ ,

where τ ∈ R, p _⊥ ∈ R ^d , p _⊥ · y = 0, I 2 ≤ 2c 6

ε ²

|y|

X

z ∈Z

Z

ξ ∈R

, | ξ |≤ r

1 + _|y| ^π |z| + |ξ| −n

dξ ≤ c 6 c 8 ε ² ,

c 8 = 2

|y|

| S ^{d − 2} | n − d + 1

X

z ∈Z

(1 + _| ^π _{y |} |z|) ^d−n−1 .

(83)

In addition, if r = r 2 (E), ε = ε 2 (E), where r 2 (E), ε 2 (E) are defined in (45), then

r ^d−n = (2τ) ^d−n E ^−α

, ε ^{− 1} E ⁻

r ^d+β = (2τ) ^d+β E ^{− α}

,

ε ² = E ^{− α}

.

(84)

Using representation (78), estimates (58), (79), (82), (83), formulas (84) and taking into account definitions (45), we obtain

|u(x, E) − v(x)| ≤ A 2 E ^−α

, x ∈ D, E

≥ ρ 2 , A 2 = A 2 (D 0 , D 1 , D 2 , N 0 , N 1 , N 2 , M, M 1 , . . . , M d , d, n, β, τ, |y|)

= (2τ) ^d+β c 7 + c 6 c 8 + (2τ) ^d−n c 3 kvk n,1 ,

(85)

where D j , N j are the same as in estimates (15) and kvk n,1 ≤ M , kx j vk n,1 ≤ M j . Theorem 2 is proved.

7 Proof of Theorem 3

Proposition 3. Let v satisfy (2) and w 1 , . . . , w d+1 be the same as in (24), (29), d ≥ 2. Then:

| b v(p) − U w

,..., w

(p, E)| ≤ c 9 ε ^{− 1} (1 + |p|) ^β E ⁻

, p ∈ B ₂ ^√ _E \ Z ^ε

w

,..., w

, E

≥ ρ 3 , 0 < ε < 1, c 9 = ^{π √} ₂ ^d 2c(D 0 )N ₀ ³ + c(D 1 )N ₁ ³ + max

j=2,...,d+1 c(D j )N _j ³ c 1 , ρ 3 = max

j=0,...,d+1 ρ(D j , N j ),

(86)

in addition, if v ∈ W ^n,1 ( R ^d ), n ≥ 0, then:

| b v(p) − U ^ε

w

,..., w

(p, E)|

≤ 2 ^β c 9 ε ⁻¹ (1 + |p|) ^β E ⁻

+ 2c 6 ε 1 + 2 ^π _s kz(p)k 2 − n

, p ∈ B ₂ ^√ _E ∩ Z ^ε

w

,..., w

, E

≥ ρ 3 , 0 < ε < min{1, ¹ ₂ s},

(87)

where c, ρ, D j , N j , j = 0, . . . , d + 1, are defined as in (15); c 1 , β are the same as in Lemma 1, c 6 is the same as in Proposition 2, z(p) is defined in (48) and kz(p)k 2 is the standard Euclidean norm of z(p).

Proof of Proposition 3. In a similar way with formulas (67), one can write M w

, w

(p) = w b 1 (p)

1 0

cos(sp i

− 1 ) sin(sp i

− 1 )

, p ∈ R ^d \ ^π _s Z ^d , M ⁻¹

w

, w

(p) = 1 sin(sp i

− 1 ) w b 1 (p)

sin(sp i

− 1 ) 0

− cos(sp i

− 1 ) 1

, p ∈ R ^d \ Z ^ε

w

,..., w

, (88)

where i ⁰ = i ⁰ (p, s) is defined in (49). Also note that

| sin(sp i

−1 )| ≥ _π ^{2ε √} _d , p ∈ R ^d \ Z ^ε

w

,..., w

, 0 < ε < 1. (89)

Estimate (86) follows from (11), (15), (23), (35), (38), (39), (49), (50) and from

(88), (89).

It remains to prove (87). Using definition (51), we represent U ^ε

w

,..., w

(p, E) − b v(p) = ϕ ^ε ₁ (p, E) + ϕ ^ε ₂ (p), p ∈ B ₂ ^√ _E ∩ Z ^ε

w

,..., w

ϕ ^ε ₁ (p, E) = 1

| S ^{d − 1} | Z

S^d−1

U w

,..., w

(η, E) − b v(η) _η= ε

s ϑ + π s z (p) dϑ, ϕ ^ε ₂ (p) = 1

| S ^d−1 | Z

S^d−1

b v ^ε _s ϑ + ^π _s z(p)

− b v(p) dϑ,

(90)

where z(p) is defined in (48).

Using formulas (86), (90), we obtain

|ϕ ^ε ₁ (p, E)| ≤ c 9 ε ^{− 1} E ⁻

1

| S ^{d − 1} | Z

S^d−1

1 +

^ε _s ϑ + ^π _s z(p)

β

dϑ

≤ c 9 ε ⁻¹ E ⁻

1

| S ^d−1 | Z

S^d−1

1 + |p| + 2 ^ε _s β

dϑ ≤ 2 ^β c 9 ε ⁻¹ (1 + |p|) ^β E ⁻

, for ε as in (87).

(91)

Next, using the definition of ϕ ^ε ₂ in formula (90) and the mean value theorem, we get the following estimate:

|ϕ ^ε ₂ (p)| ≤ 2 ^ε _s max

|∇ b v(ξ)|: ξ ∈ R ^d , |ξ − ^π _s z(p)| ≤ ^ε _s , p ∈ Z ^ε

w

,..., w

. (92) Here, the mean value theorem was used for v(ξ) on [p, b ^ε _s ϑ + ^π _s z(p)], ϑ ∈ S ^d−1 . One can see that

estimates (72) and (73) hold for all ξ ∈ R ^d such that |ξ − ^π _s z(p)| ≤ ^ε _s , where p ∈ Z ^ε

w

,..., w

. (93) It follows from (92), (93) and from the upper estimate on ε of (87), that

|ϕ ^ε ₂ (p)| ≤ 2 ¹⁻ⁿ c 6 ε max

(1 + |ξ|) ⁻ⁿ : |ξ − ^π _s z(p)| ≤ ^ε _s

≤ 2 ¹⁻ⁿ c 6 ε 1 + ^π _s kz(p)k 2 − ^ε _s −n

≤ 2c 6 ε 1 + 2 ^π _s kz(p)k 2 −n

, p ∈ Z ^ε

w

,..., w

.

(94)

Estimate (87) follows from estimates (91) and (94).

Proposition 3 is proved.

The final part of the proof of Theorem 3 is as follows. In a similar way with (78), we represent v as follows:

v(x) = v ⁺ ₁ (x, r) + v ⁺ ₂ (x, r) + v ⁻ (x, r), x ∈ D, r > 0, v ⁺ ₁ (x, r) =

Z

B

\ Z

wb1,...,wd+1b

e ^{− ipx} b v(p) dp, v ⁺ ₂ (x, r) =

Z

B

∩Z

wb1,...,wd+1b

e ^−ipx b v(p) dp, v ⁻ (x, r) =

Z

R^d

\B

e ^{− ipx} b v(p) dp.

(95)

Since v belongs to W ^n,1 (R ^d ), estimate (58) is valid.

Using estimates (86), (87) we obtain

v ₁ ⁺ (x, r) − Z

B

\ Z

wb1,...,wd+1b

e ^−ipx U

w

,..., w

(p, E) dp +v ₂ ⁺ (x, r) −

Z

B

∩ Z

wb1,...,wd+1b

e ^−ipx U ^ε

w

,..., w

(p, E) dp

≤ J 1 + J 2 , J 1 = 2 ^β c 9 ε ^{− 1} E ⁻

Z

B

(1 + |p|) ^β dp, J 2 = 2c 6 ε

Z

B

∩ Z

wb1,...,wd+1b

1 + 2 ^π _s kz(p)k 2 − n

dp, x ∈ D, 1 ≤ r ≤ 2E

, E

≥ ρ 3 ,

(96)

where ρ 3 is the same as in Proposition 3. In addition, J 1 ≤ c 10 ε ⁻¹ E ⁻

r ^d+β , c 10 = | S ^d−1 | ² _d+β

c 9 , J 2 ≤ c 11 ε ^d+1 , c 11 = ¹ _s d

|B 1 | X

z ∈Z

1 + 2 ^π _s kzk 2 − n

, (97)

where |B ₁ | is the standard Euclidean volume of B ₁ . Finally, if r = r 3 (E), ε = ε 3 (E), where r 3 (E), ε 3 (E) are defined in (53), then

r ^{d − n} = (2τ) ^{d − n} E ^{− α}

, ε ^{− 1} E ⁻

r ^d+β = (2τ) ^d+β E ^{− α}

,

ε ^d+1 = E ^{− α}

.

(98)

Using representation (95), estimates (58), (96), (97), formulas (98) and taking into account definitions (53), we obtain

|u(x, E) − v(x)| ≤ A 3 E ^−α

, x ∈ D, E

≥ ρ 3 , A 3 = A 3 (D 0 , . . . , D d+1 , N 0 , . . . , N d+1 , M, d, n, β, τ, s)

= (2τ) ^d+β c 10 + c 11 + (2τ ) ^{d − n} c 3 kvk n,1 ,

(99)

where kvk n,1 ≤ M and D 0 , . . . , D d+1 , N 0 , . . . , N d+1 are the same as in Propo- sition 3.

Theorem 3 is proved.

8 Proof of Lemma 1

Note that

w(p) = b Z

R^d

| b q(ξ)| ² b ω ν (p − ξ) dξ, p ∈ R ^d , (100) ω ν (x) = |x| ^ν K ν |x|

, x ∈ R ^d , (101)

where q, b ω b ν are the Fourier transforms of q, ω ν . The Fourier transform ω b ν can be computed explicitely:

ω b ν (p) = c 12

(1 + |p| ² )

^+ν , c 12 = Γ( ^d ₂ + ν)2 ^{ν − 1}

π

. (102)

Indeed, formula (102) follows from the Fourier inversion theorem and the fol- lowing computations:

Z

R^d

e ^{− ipx} dp (1 + |p| ² )

^+ν =

Z

R

Z

R^d−1

e ^{− i | x | t} dt dξ (1 + t ² + |ξ| ² )

^+ν

= | S ^{d − 2} | Z

R

Z

R

e ^{− i | x | t} r ^{d − 2} dt dr (1 + t ² + r ² )

^+ν

r= √ 1+t

τ

====== | S ^{d − 2} | Z

R

e ^{− i | x | t} dt (1 + t ² )

^+ν

Z +∞

0

τ ^{d − 2} dτ (1 + τ ² )

^+ν

= c ⁻ ₁₂ ¹ |x| ^ν K ν (|x|), x ∈ R ^d . Here, it was used that

Z + ∞ 0

τ ^{d − 2} dτ (1 + τ ² )

^+ν = 1

2 B( ^d−1 ₂ , ν + ¹ ₂ ) = 1 2

Γ( ^d−1 ₂ )Γ(ν + ¹ ₂ ) Γ( ^d ₂ + ν) ,

| S ^{d − 2} | = 2π

Γ( ^{d −} ₂ ¹ ) , where B and Γ denote the beta and gamma functions.

Using (100), (102), we obtain the estimates

w(p) b ≥ Z

| ξ |≤ 1

c 12 | q(ξ)| b ²

(1 + |p − ξ|) ^d+2ν dξ ≥ c 1 (q, ν )

(1 + |p|) ^d+2ν , p ∈ R ^d , c 1 (q, ν) = c 12

2 ^d+2ν Z

| ξ |≤ 1

| q(ξ)| b ² dξ.

(103)

Properties (23) follow from (20), (22), (100) and (103).

Lemma 1 is proved.

Aknowledgements

The authors are grateful to the referee for remarks that have helped to improve the presentation.

References

[1] A. D. Agaltsov, T. Hohage, and R. G. Novikov, “An iterative approach to

monochromatic phaseless inverse scattering,” in preparation.

[2] T. Aktosun and P. E. Sacks, “Inverse problems on the line without phase information,” Inverse Problems, vol. 14, pp. 211–224, 1998.

[3] N. V. Alexeenko, V. A. Burov, and O. D. Rumyantseva, “Solution of the three-dimensional acoustical inverse scattering problem. The modified Novikov algorithm,” Acoustical Physics , vol. 54, no. 3, pp. 407–419, 2008.

[4] J. A. Barcel´ o, C. Castro, and J. M. Reyes, “Numerical approximation of the potential in the two-dimesional inverse scattering problem,” Inverse Problems, vol. 32, no. 1, 2016, 015006 (19pp).

[5] F. A. Berezin and M. A. Shubin, The Schr¨ odinger equation, ser. Mathe- matics and its applications. Dordrecht: Kluwer Academic, 1991, vol. 66.

[6] V. A. Burov, N. V. Alekseenko, and O. D. Rumyantseva, “Multifrequency generalization of the Novikov algorithm for the two-dimensional inverse scattering problem,” Acoustical Physics, vol. 56, no. 6, pp. 843–856, 2009.

[7] K. Chadan and P. C. Sabatier, Inverse problems in quantum scattering theory, 2nd ed. Berlin: Springer, 1989.

[8] G. Eskin, Lectures on linear partial differential equations , ser. Graduate studies in mathematics. American mathematical society, 2011, vol. 123.

[9] L. D. Faddeev, “Uniqueness of the solution of the inverse scattering prob- lem,” Vest. Leningrad Univ., vol. 7, pp. 126–130, 1956, (in Russian).

[10] ——, “Inverse problem of quantum scattering theory. II.” Journal of Soviet Mathematics, vol. 5, no. 3, pp. 334–396, 1976.

[11] L. D. Faddeev and S. P. Merkuriev, Quantum scattering theory for sev- eral particle systems, ser. Mathematical physics and applied mathematics.

Dordrecht: Kluwer Academic, 1993, vol. 11.

[12] A. A. Gonchar, G. M. Henkin, and N. N. Novikova, “Multipoint Pad´e approximants in an inverse Sturm-Liouville problem,” Math. USSR-Sb., vol. 73, no. 2, pp. 479–489, 1992.

[13] P. G. Grinevich, “The scattering transform for the two-dimensional Schr¨odinger operator with a potential that decreases at infinity at fixed nonzero energy,” Russian Math. Surveys, vol. 55, no. 6, pp. 1015–1083, 2000.

[14] P. H¨ahner and T. Hohage, “New stability estimates for the inverse acoustic inhomogeneous medium problem and applications,” SIAM J. Math. Anal., vol. 33, no. 3, pp. 670–685, 2001.

[15] G. M. Henkin and R. G. Novikov, “The ¯ ∂-equation in the multidimensional

inverse scattering problem,” Russian Mathematical Surveys, vol. 42, no. 3,

pp. 109–180, 1987.

[16] ——, “A multidimensional inverse problem in quantum and acoustic scat- tering,” Inv. Problems, vol. 4, pp. 103–121, 1988.

[17] M. I. Isayev, “Exponential instability in the inverse scattering problem on the energy interval,” Funct. Anal. Appl., vol. 47, no. 3, pp. 187–194, 2013.

[18] M. I. Isayev and R. G. Novikov, “New global stability estimates for monochromatic inverse acoustic scattering,” SIAM J. Math. Analysis , vol. 43, no. 5, pp. 1495–1504, 2013.

[19] M. V. Klibanov, “Phaseless inverse scattering problems in three dimen- sions,” SIAM J. Appl. Math., vol. 74, pp. 392–410, 2014.

[20] M. V. Klibanov and V. G. Romanov, “Reconstruction procedures for two inverse scattering problems without the phase information,” SIAM J. Appl.

Math., vol. 76, no. 1, pp. 178–196, 2016.

[21] ——, “Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation,” Inverse Problems, vol. 32, no. 1, 2016, 015005 (16pp).

[22] R. G. Novikov, “Multidimensional inverse spectral problem for the equation

−∆ψ + (v(x) − Eu(x))ψ = 0,” Funct. Anal. Appl. , vol. 22, pp. 263–272, 1988.

[23] ——, “Rapidly converging approximation in inverse quantum scattering in dimension 2,” Physics Letters A, vol. 238, pp. 73–78, 1998.

[24] ——, “The ¯ ∂-approach to approximate inverse scattering at fixed energy in three dimensions,” Int. Math. Res. Pap., no. 6, pp. 287–349, 2005.

[25] ——, “Approximate Lipschitz stability for non-overdetermined inverse scat- tering at fixed energy,” J. Inverse Ill-Posed Probl., vol. 26, no. 6, pp. 813–

823, 2013.

[26] ——, “Inverse scattering without phase information,” in S´ eminaire Laurent Schwartz — EDP et applications. Ecole Polytechnique, 2014-2015, exp.

no. 16 (13pp).

[27] ——, “Formulas for phase recovering from phaseless scattering data at fixed frequency,” Bull. Sci. Math., vol. 139, no. 8, pp. 923–936, 2015.

[28] ——, “An iterative approach to non-overdetermined inverse scattering at fixed energy,” Sbornik: Mathematics, vol. 206, no. 1, pp. 120–134, 2015.

[29] ——, “Explicit formulas and global uniqueness for phaseless inverse scat-

tering in multidimensions,” The Journal of Geometric Analysis, vol. 26,

no. 1, pp. 346–359, 2016.