# Energy- and Regularity-Dependent Stability Estimates for Near-Field Inverse Scattering in Multidimensions

Authors: M. I. Isaev
Journal: Journal of Mathematics
 (2013)

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## Abstract

We prove new global Hölder-logarithmic stability estimates for the near-field  inverse scattering  problem in dimensiond≥3. Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. In addition, a global logarithmic stability estimate for this inverse problem in dimension d=2 is also given.


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## Body

## 1. Introduction

We consider the Schrödinger equation:(1)Lψ=Eψ,L=-Δ+v(x),x∈ℝd,d≥2,
where
(2)visreal-valued,v∈𝕃∞(ℝd),v(x)=O(|x|-d-ε),|x|→∞,forsomeε>0.
We consider the resolvent R(E) of the Schrödinger operator L in 𝕃2(ℝd):
(3)R(E)=(L-E)-1,E∈ℂ∖σ(L),
where σ(L) is the spectrum of L in 𝕃2(ℝd). We assume that R(x,y,E) denotes the Schwartz kernel of R(E) as of an integral operator. We consider also
(4)R+(x,y,E)=R(x,y,E+i0),x,y∈ℝd,E∈ℝ+.
We recall that in the framework of (1) the function R+(x,y,E) describes scattering of the spherical waves,
(5)R0+(x,y,E)=-i4(E2π|x-y|)(d-2)/2H(d-2)/2(1)(E|x-y|),
generated by a source at y (where Hμ(1) is the Hankel function of the first kind of order μ). We recall also that R+(x,y,E) is the Green function for L-E, E∈ℝ+, with the Sommerfeld radiation condition at infinity.In addition, the function(6)S+(x,y,E)=R+(x,y,E)-R0+(x,y,E),x,y∈∂Br,E∈ℝ+,r∈ℝ+
is considered as near-field scattering data for (1), where Br is the open ball of radius r centered at 0.We consider, in particular, the following near-field inverse scattering problem for (1).Problem 1.
GivenS+ on ∂Br×∂Br for some fixed r, E∈ℝ+, find v on Br.This problem can be considered under the assumption thatv is a priori known on ℝd∖Br. Actually, in the present paper we consider Problem 1 under the assumption that v≡0 on ℝd∖Br for some fixed r∈ℝ+. Below in this paper we always assume that this additional condition is fulfilled.It is well known that the near-field scattering data of Problem1 uniquely and efficiently determine the scattering amplitude f for (1) at fixed energy E, see [1]. Therefore, approaches of [2–12] can be applied to Problem 1 via this reduction.In addition, it is also known that the near-field data of Problem1 uniquely determine the Dirichlet-to-Neumann map in the case when E is not a Dirichlet eigenvalue for operator L in Br, see [8, 13]. Therefore, approaches of [3, 8, 14–24] can be also applied to Problem 1 via this reduction.However, in some case it is much more optimal to deal with Problem1 directly, see, for example, logarithmic stability results of [25] for Problem 1 in dimension d=3. A principal improvement of estimates of [25] was given recently in [26]: stability of [26] efficiently increases with increasing regularity of v.Problem1 can be also considered as an example of ill-posed problem: see [27, 28] for an introduction to this theory.In the present paper we continue studies of [25, 26]. We give new global Hölder-logarithmic stability estimates for Problem 1 in dimension d≥3, see Theorem 1. Our estimates are given in uniform norm for coefficient difference and related stability efficiently increases with increasing energy and/or coefficient regularity. Results of such a type for the Gel'fand inverse problem were obtained recently in [15] for d≥3 and in [29] for d=2.The main feature of our new estimates is the explicit dependence on the energyE. These estimates consist of two parts, the first is Hölder and the second is logarithmic; when E increases, the logarithmic part decreases and the Hölder part becomes dominant.In addition, we give also global logarithmic stability estimates for Problem1 in dimension d=2, see Theorem 2.


## 2. Stability Estimates

We recall that ifv satisfies (2) and suppv⊂Br1 for some r1>0, then
(7)S+(E)isboundedin𝕃2(∂Br×∂Br)foranyr>r1,
where S+(E) is the near-field scattering data of v for (1) with E>0, for more details see, for example, Section 2 of [25].


### 2.1. Estimates ford≥3

In this subsection we assume for simplicity that(8)v∈𝕎m,1(ℝd)forsomem>d,visreal-valued,suppv⊂Br1forsomer1>0,
where
(9)𝕎m,1(ℝd)={v:∂Jv∈𝕃1(ℝd),|J|≤m},m∈ℕ∪0,
where
(10)J∈(ℕ∪0)d,|J|=∑i=1d‍Ji,∂Jv(x)=∂|J|v(x)∂x1J1⋯∂xdJd.
Let
(11)∥v∥m,1=max|J|≤m∥∂Jv∥𝕃1(ℝd).Note that (8)⇒(2).Theorem 1.
LetE>0 and r>r1 be given constants. Let dimension d≥3 and potentials v1, v2 satisfy (8). Let ∥vj∥m,1≤N,j=1,2, for some N>0. Let S1+(E) and S2+(E) denote the near-field scattering data for v1 and v2, respectively. Then for τ∈(0,1) and any s∈[0,s*] the following estimate holds:
(12)∥v2-v1∥L∞(Br)≤C1(1+E)5/2δτ+C2(1+E)(s-s*)/2(ln(3+δ-1))-s,
where s*=(m-d)/d, δ=∥S1+(E)-S2+(E)∥𝕃2(∂Br×∂Br), and constants C1,C2>0 depend only on N,m,d,r,τ.Proof of Theorem1 is given in Section 5. This proof is based on results presented in Sections 3 and 4.


### 2.2. Estimates ford=2

In this subsection we assume for simplicity that(13)visreal-valued,v∈C2(B¯r1),suppv⊂Br1forsomer1>0.
Note also that (13)⇒(2).Theorem 2.
LetE>0 and r>r1 be given constants. Let dimension d=2 and potentials v1, v2 satisfy (13). Let ∥vj∥C2(Br)≤N,j=1,2, for some N>0. Let S1+(E) and S2+(E) denote the near-field scattering data for v1 and v2, respectively. Then
(14)∥v1-v2∥L∞(Br)≤C3(ln(3+δ-1))-3/4(ln(3ln(3+δ-1)))2,
where δ=∥S1+(E)-S2+(E)∥𝕃2(∂Br×∂Br) and constant C3>0 depends only on N, m, r.Proof of Theorem2 is given in Section 7. This proof is based on results presented in Sections 3 and 6.


### 2.3. Concluding Remarks

Remark 3.
The logarithmic stability estimates for Problem 1.1 of [25, 26] follow from estimate (12) for d=3 and s=s*. Apparently, using the methods of [19, 20] it is possible to improve estimate (12) for s*=m-d.Remark 4.
In the same way as in [25, 26] for dimesnsion d=3, using estimates (12) and (14), one can obtain logarithmic stability estimates for the reconstruction of a potential v from the inverse scattering amplitude f for any d≥2.Remark 5.
Actually, in the proof of Theorem1 we obtain the following estimate (see formula (57)):
(15)∥v1-v2∥𝕃∞(Br)≤C4(1+E)2E+ρ2e2ρ(r+1)δ+C5(E+ρ2)-(m-d)/2d,
where constants C4,C5>0 depend only on N,m,d,r and the parameter ρ>0 is such that E+ρ2 is sufficiently large: E+ρ2≥C6(N,r,m). Estimate of Theorem 1 follows from estimate (15).


## 2.1. Estimates ford≥3

In this subsection we assume for simplicity that(8)v∈𝕎m,1(ℝd)forsomem>d,visreal-valued,suppv⊂Br1forsomer1>0,
where
(9)𝕎m,1(ℝd)={v:∂Jv∈𝕃1(ℝd),|J|≤m},m∈ℕ∪0,
where
(10)J∈(ℕ∪0)d,|J|=∑i=1d‍Ji,∂Jv(x)=∂|J|v(x)∂x1J1⋯∂xdJd.
Let
(11)∥v∥m,1=max|J|≤m∥∂Jv∥𝕃1(ℝd).Note that (8)⇒(2).Theorem 1.
LetE>0 and r>r1 be given constants. Let dimension d≥3 and potentials v1, v2 satisfy (8). Let ∥vj∥m,1≤N,j=1,2, for some N>0. Let S1+(E) and S2+(E) denote the near-field scattering data for v1 and v2, respectively. Then for τ∈(0,1) and any s∈[0,s*] the following estimate holds:
(12)∥v2-v1∥L∞(Br)≤C1(1+E)5/2δτ+C2(1+E)(s-s*)/2(ln(3+δ-1))-s,
where s*=(m-d)/d, δ=∥S1+(E)-S2+(E)∥𝕃2(∂Br×∂Br), and constants C1,C2>0 depend only on N,m,d,r,τ.Proof of Theorem1 is given in Section 5. This proof is based on results presented in Sections 3 and 4.


## 2.2. Estimates ford=2

In this subsection we assume for simplicity that(13)visreal-valued,v∈C2(B¯r1),suppv⊂Br1forsomer1>0.
Note also that (13)⇒(2).Theorem 2.
LetE>0 and r>r1 be given constants. Let dimension d=2 and potentials v1, v2 satisfy (13). Let ∥vj∥C2(Br)≤N,j=1,2, for some N>0. Let S1+(E) and S2+(E) denote the near-field scattering data for v1 and v2, respectively. Then
(14)∥v1-v2∥L∞(Br)≤C3(ln(3+δ-1))-3/4(ln(3ln(3+δ-1)))2,
where δ=∥S1+(E)-S2+(E)∥𝕃2(∂Br×∂Br) and constant C3>0 depends only on N, m, r.Proof of Theorem2 is given in Section 7. This proof is based on results presented in Sections 3 and 6.


## 2.3. Concluding Remarks

Remark 3.
The logarithmic stability estimates for Problem 1.1 of [25, 26] follow from estimate (12) for d=3 and s=s*. Apparently, using the methods of [19, 20] it is possible to improve estimate (12) for s*=m-d.Remark 4.
In the same way as in [25, 26] for dimesnsion d=3, using estimates (12) and (14), one can obtain logarithmic stability estimates for the reconstruction of a potential v from the inverse scattering amplitude f for any d≥2.Remark 5.
Actually, in the proof of Theorem1 we obtain the following estimate (see formula (57)):
(15)∥v1-v2∥𝕃∞(Br)≤C4(1+E)2E+ρ2e2ρ(r+1)δ+C5(E+ρ2)-(m-d)/2d,
where constants C4,C5>0 depend only on N,m,d,r and the parameter ρ>0 is such that E+ρ2 is sufficiently large: E+ρ2≥C6(N,r,m). Estimate of Theorem 1 follows from estimate (15).


## 3. Alessandrini-Type Identity for Near-Field Scattering

In this section we always assume that assumptions of Theorems1 and 2 are fulfilled (in the cases of dimension d≥3 and d=2, resp.).Consider the operatorsR^j,j=1,2, defined as follows
(16)(R^jϕ)(x)=∫∂Br‍Rj+(x,y,E)ϕ(y)dy,x∈∂Br,j=1,2.
Note that
(17)∥R^1-R^2∥𝕃2(∂Br)≤∥S1+(E)-S2+(E)∥𝕃2(∂Br)×𝕃2(∂Br).
We recall that (see [25]) for any functions ϕ1,ϕ2∈C(ℝd), sufficiently regular in ℝd∖∂Br and satisfying
(18)-Δϕ+v(x)ϕ=Eϕ,inℝd∖∂Br,lim|x|→+∞|x|(d-1)/2(∂∂|x|ϕ-iEϕ)=0,
with v=v1 and v=v2, respectively, the following identity holds:
(19)∫Br‍(v2-v1)ϕ1ϕ2dx=∫∂Br‍(∂ϕ1∂ν+-∂ϕ1∂ν-)[(R^1-R^2)(∂ϕ2∂ν+-∂ϕ2∂ν-)]dx,
where where ν+ and ν- are the outward and inward normals to ∂Br, respectively.Remark 6.
The identity (19) is similar to the Alessandrini identity (see Lemma 1 of [14]), where the Dirichlet-to-Neumann maps are considered instead of operators R^j.To apply identity (19) to our considerations, we use also the following lemma.Lemma 7.
LetE,r>0 and d≥2. Then, there is a positive constant C7 (depending only on r and d) such that for any ϕ∈C(ℝd∖Br) satisfying
(20)-Δϕ=Eϕ,inℝd∖B¯r,lim|x|→+∞|x|(d-1)/2(∂∂|x|ϕ-iEϕ)=0,ϕ|∂Br∈ℍ1(∂Br),
the following inequality holds:
(21)∥∂ϕ∂ν+|∂Br∥𝕃2(∂Br)≤C7(1+E)∥ϕ|∂Br∥ℍ1(∂Br),
where ℍ1(∂Br) denotes the standard Sobolev space on ∂Br.The proof of Lemma7 is given in Section 8.


## 4. Faddeev Functions

In dimensiond≥3, we consider the Faddeev functions h, ψ, G (see [6, 8, 30, 31]):
(22)h(k,l)=(2π)-d∫ℝd‍e-ilxv(x)ψ(x,k)dx,
where k,l∈ℂd, k2=l2, Imk=Iml≠0,
(23)ψ(x,k)=eikx+∫ℝd‍G(x-y,k)v(y)ψ(y,k)dy,(24)G(x,k)=eikxg(x,k),g(x,k)=-(2π)-d∫ℝd‍eiξxdξξ2+2kξ,
where x∈ℝd, k∈ℂd, Imk≠0, d≥3,One can consider (22), (23) assuming that
(25)visasufficientlyregularfunctiononℝdwithsuffucientdecayatinfinity.
For example, in connection with Theorem 1, we consider (22), (23) assuming that
(26)v∈𝕃∞(Br),v≡0onℝ∖Br.We recall that (see [6, 8, 30, 31])
(27)(Δ+k2)G(x,k)=δ(x),x∈ℝd,k∈ℂd∖ℝd,
formula (23) at fixed k is considered as an equation for
(28)ψ=eikxμ(x,k),
where μ is sought in 𝕃∞(ℝd); as a corollary of (23), (24), and (27), ψ satisfies (1) for E=k2; h of (22) is a generalized “scattering” amplitude.In addition,h, ψ, G in their zero energy restriction, that is for E=0 were considered for the first time in [32]. The Faddeev functions h, ψ, G were, actually, rediscovered in [32].Let(29)ΣE={k∈ℂd:k2=k12+⋯+kd2=E},ΘE={k∈ΣE,l∈ΣE:Imk=Iml},|k|=(|Rek|2+|Imk|2)1/2.
Let
(30)vsatisfy(8),∥v∥m,1≤N,v^(p)=(2π)-d∫ℝd‍eipxv(x)dx,p∈ℝd,
then we have that
(31)μ(x,k)→1as|k|→∞
and, for any σ>1,
(32)|μ(x,k)|+|∇μ(x,k)|≤σfor|k|≥λ1(N,m,d,r,σ),
where x∈ℝd, k∈ΣE;
(33)v^(p)=lim(k,l)∈ΘE,k-l=p|Imk|=|Iml|→∞h(k,l),foranyp∈ℝd,(34)|v^(p)-h(k,l)|≤c1(m,d,r)N2(E+ρ2)1/2,for(k,l)∈ΘE,p=k-l,|Imk|=|Iml|=ρ,E+ρ2≥λ2(N,m,d,r),p2≤4(E+ρ2).Results of the type (31), (32) go back to [32]. For more information concerning (32) see estimate (4.11) of [33]. Results of the type (33), (34) (with less precise right-hand side in (34)) go back to [6]. Estimate (34) follows, for example, from formulas (23), (22), and the estimate
(35)∥Λ-sg(k)Λ-s∥𝕃2(ℝd)→𝕃2(ℝd)=O(|k|-1)as|k|→∞,k∈ℂd∖ℝd,
for s>1/2, where g(k) denotes the integral operator with the Schwartz kernel g(x-y,k) and Λ denotes the multiplication operator by the function (1+|x|2)1/2. Estimate (35) was formulated, first, in [34] for d≥3. Concerning proof of (35), see [35].In addition, we have that(36)h2(k,l)-h1(k,l)=(2π)-d∫ℝd‍ψ1(x,-l)(v2(x)-v1(x))ψ2(x,k)dxfor(k,l)∈ΘE,|Imk|=|Iml|≠0,andv1,v2satisfying(25),
and, under assumptions of Theorem 1,
(37)|v^1(p)-v^2(p)-h1(k,l)+h2(k,l)|≤c2(m,d,r)N∥v1-v2∥𝕃∞(Br)(E+ρ2)1/2for(k,l)∈ΘE,p=k-l,|Imk|=|Iml|=ρ,E+ρ2≥λ3(N,m,d,r),p2≤4(E+ρ2),
where hj, ψj denote h and ψ of (22) and (23) for v=vj,j=1,2.Formula (36) was given in [36]. Estimate (37) was given for example in [15].


## 5. Proof of Theorem1

Let(38)𝕃μ∞(ℝd)={u∈𝕃∞(ℝd):∥u∥μ<+∞},∥u∥μ=esssupp∈ℝd(1+|p|)μ|u(p)|,μ>0.Note that(39)w∈𝕎m,1(ℝd)⇒w^∈𝕃μ∞(ℝd)∩C(ℝd),∥w^∥μ≤c3(m,d)∥w∥m,1,forμ=m,
where 𝕎m,1, 𝕃μ∞ are the spaces of (9), (38),
(40)w^(p)=(2π)-d∫ℝd‍eipxw(x)dx,p∈ℝd.Using the inverse Fourier transform formula(41)w(x)=∫ℝd‍e-ipxw^(p)dp,x∈ℝd,
we have that
(42)∥v1-v2∥𝕃∞(Br)≤supx∈B¯r|∫ℝd‍e-ipx(v^2(p)-v^1(p))dp|≤I1(κ)+I2(κ),foranyκ>0,
where
(43)I1(κ)=∫|p|≤κ‍|v^2(p)-v^1(p)|dp,I2(κ)=∫|p|≥κ‍|v^2(p)-v^1(p)|dp.Using (39), we obtain that
(44)|v^2(p)-v^1(p)|≤2c3(m,d)N(1+|p|)-m,p∈ℝd.
Let
(45)c4=∫p∈ℝd,|p|=1‍dp.
Combining (43), (44), we find that, for any κ>0,
(46)I2(κ)≤2c3(m,d)Nc4∫κ+∞‍dttm-d+1≤2c3(m,d)Nc4m-d1κm-d.Due to (37), we have that
(47)|v^2(p)-v^1(p)|≤|h2(k,l)-h1(k,l)|+c2(m,d,r)N∥v1-v2∥𝕃∞(Br)(E+ρ2)1/2,for(k,l)∈ΘE,p=k-l,|Imk|=|Iml|=ρ,E+ρ2≥λ3(N,m,d,r),p2≤4(E+ρ2).Let(48)δ=∥S1+(E)-S2+(E)∥𝕃2(∂Br×∂Br).
Combining (17), (19), and (36), we get that
(49)|h2(k,l)-h1(k,l)|≤δ∥∂ϕ1∂ν+-∂ϕ1∂ν-∥𝕃2(Br)∥∂ϕ2∂ν+-∂ϕ2∂ν-∥𝕃2(Br),(k,l)∈ΘE,|Imk|=|Iml|≠0,
where ϕj, j=1,2, denotes the solution of (18) with v=vj, satisfying
(50)ϕj(x)=ψj(x,k),forx∈B¯r.
Using (21), (32), and the fact that C1(∂Br)⊂ℍ1(∂Br), we find that
(51)∥∂ϕj∂ν+-∂ϕj∂ν-∥𝕃2(Br)≤σc5(r,d)(1+E)exp(|Imk|(r+1)),k∈ΣE,|k|≥λ1(N,m,d,r,σ),j=1,2.
Here and bellow in this section the constant σ is the same that in (32).Combining (49) and (51), we obtain that
(52)|h2(k,l)-h1(k,l)|≤c52σ2(1+E)2e2ρ(r+1)δ,for(k,l)∈ΘE,ρ=|Imk|=|Iml|,E+ρ2≥λ12(N,m,d,r,σ).
Using (47), (52), we get that
(53)|v^2(p)-v^1(p)|≤c52σ2(1+E)2e2ρ(r+1)δ+c2(m,d,r)N∥v1-v2∥𝕃∞(B1)(E+ρ2)1/2,p∈ℝd,p2≤4(E+ρ2),E+ρ2≥max{λ12,λ3}.
Let
(54)ε=(12c2(m,d,r)Nc6)1/d,c6=∫p∈ℝd,|p|≤1‍dp,
and λ4(N,m,d,r,σ)>0 be such that
(55)E+ρ2≥λ4(N,m,d,r,σ)⇒{E+ρ2≥λ12(N,m,d,r,σ),E+ρ2≥λ3(N,m,d,r),(ε(E+ρ2)1/2d)2≤4(E+ρ2).
Using (43), (53), we get that
(56)I1(κ)≤c6κd(112c52σ2(1+E)2e2ρ(r+1)δ+c2(m,d,r)N∥v1-v2∥𝕃∞(B1)(E+ρ2)1/2),κ>0,κ2≤4(E+ρ2),E+ρ2≥λ4(N,m,d,r,σ).
Combining (42), (46), and (56) for κ=ε(E+ρ2)1/2d and (55), we get that
(57)∥v1-v2∥𝕃∞(Br)≤c7(N,m,d,r,σ)(1+E)2E+ρ2e2ρ(r+1)δ+c8(N,m,d)(E+ρ2)-(m-d)/2d+12∥v1-v2∥𝕃∞(Br),E+ρ2≥λ4(N,m,d,r,σ).Letτ′∈(0,1),
(58)β=1-τ′2(r+1),ρ=βln(3+δ-1),
and δ1=δ1(N,m,d,σ,r,τ′)>0 be such that
(59)δ∈(0,δ1)⇒{E+(βln(3+δ-1))2≥λ4(N,m,d,r,σ),E+(βln(3+δ-1))2≤(1+E)(βln(3+δ-1))2.
Then for the case when δ∈(0,δ1), due to (57), we have that
(60)12∥v1-v2∥𝕃∞(Br)≤c7(1+E)2(E+(βln(3+δ-1))2)1/2(3+δ-1)2β(r+1)δ+c8(E+(βln(3+δ-1))2)-(m-d)/2d=c7(1+E)2(E+(βln(3+δ-1))2)1/2(1+3δ)1-τ′δτ′+c8(E+(βln(3+δ-1))2)-(m-d)/2d.
Combining (59) and (60), we obtain that for s∈[0,s*],τ∈(0,τ′) and δ∈(0,δ1) the following estimate holds:
(61)∥v2-v1∥L∞(Br)≤c9(1+E)5/2δτ+c10(1+E)(s-s*)/2(ln(3+δ-1))-s,
where s*=(m-d)/d and c9,c10>0 depend only on N, m, d, r, σ, τ′, and τ.Estimate (61) in the general case (with modified c9 and c10) follows from (61) for δ≤δ1(N,m,d,σ,r,τ′) and the property that
(62)∥vj∥𝕃∞(Br)≤c11(m,d)N.This completes the proof of (12).


## 6. Buckhgeim-Type Analogs of the Faddeev Functions

Let us identifyℝ2 with ℂ and use coordinates z=x1+ix2, z-=x1-ix2, where (x1,x2)∈ℝ2. Following [21–23, 37], we consider the functions Gz0, ψz0, ψ~z0, δhz0 going back to Buckhgeim’s paper [3] and being analogs of the Faddeev functions:
(63)ψz0(z,λ)=eλ(z-z0)2+∫Br‍Gz0(z,ζ,λ)v(ζ)ψz0(ζ,λ)dReζdImζ,ψ~z0(z,λ)=eλ-(z--z-0)2+∫Br‍Gz0(z,ζ,λ)¯v(ζ)ψ~z0(ζ,λ)dReζdImζ,(64)Gz0(z,ζ,λ)=14π2∫Br‍e-λ(η-z0)2+λ-(η--z-0)2dReηdImη(z-η)(η--ζ-)×eλ(z-z0)2-λ-(ζ--z-0)2,z=x1+ix2,z0∈Br,λ∈ℂ,
where v satisfies (13),
(65)δhz0(λ)=∫Br‍ψ~z0,1(z,-λ)(v2(z)-v1(z))×ψz0,2(z,λ)dRezdImz,λ∈ℂ,
where v1, v2 satisfy (13) and ψ~z0,1, ψz0,2 denote ψ~z0, ψz0 of (63) for v=v1 and v=v2, respectively.We recall that (see [21, 22]): (i)
the functionGz0 satisfies the equations
(66)4∂2∂z∂z-Gz0(z,ζ,λ)=δ(z-ζ),4∂2∂ζ∂ζ-Gz0(z,ζ,λ)=δ(z-ζ),where z,z0,ζ∈Br, λ∈ℂ, and δ is the Dirac delta function;(ii)
formulas (63) at fixed z0 and λ are considered as equations for ψz0, ψ~z0 in L∞(Br);(iii)
as a corollary of (63), (64), (66), the functions ψz0, ψ~z0 satisfy (1) in Br for E=0 and d=2;(iv)
the functionδhz0 is similar to the right side of (36).Let potentialsv,v1,v2∈C2(B¯r), and
(67)∥v∥C2(B¯r)≤N,∥vj∥C2(B¯r)≤N,j=1,2,(v1-v2)|∂Br=0,∂∂ν(v1-v2)|∂Br=0,
then we have that
(68)ψz0(z,λ)=eλ(z-z0)2μz0(z,λ),ψ~z0(z,λ)=eλ-(z--z-0)2μ~z0(z,λ),(69)μz0(z,λ)→1,μ~z0(z,λ)→1as|λ|→∞
and, for any σ>1,  (70a)|μz0(z,λ)|+|∇μz0(z,λ)|≤σ,(70b)|μ~z0(z,λ)|+|∇μ~z0(z,λ)|≤σ,where ∇=(∂/∂x1,∂/∂x2), z=x1+ix2, z0∈Br, λ∈ℂ, |λ|≥ρ1(N,r,σ);
(71)v2(z0)-v1(z0)=limλ→∞2π|λ|δhz0(λ)foranyz0∈Br,(72)|v2(z0)-v1(z0)-2π|λ|δhz0(λ)|≤c12(N,r)(ln(3|λ|))2|λ|3/4,forz0∈Br,|λ|≥ρ2(N,r).
Formulas (68) can be considered as definitions of μz0, μ~z0. Formulas (69), (71) were given in [21, 22] and go back to [3]. Estimates (70a) and (70b) were proved in [33]. Estimate (72) was obtained in [21, 37].


## 7. Proof of Theorem2

We suppose thatψ~z0,1(·,-λ),ψz0,2(·,λ),δhz0(λ) are defined as in Section 6 but with vj-E in place of vj, j=1,2. Note that functions  ψ~z0,1(·,-λ),ψz0,2(·,λ) satisfy (1) in Br with v=vj, j=1,2, respectively. We also use the notation NE=N+E. Then, using (72), we have that
(73)|v2(z0)-v1(z0)-2π|λ|δhz0(λ)|≤c12(NE,r)(ln(3|λ|))2|λ|3/4,forz0∈Br,|λ|≥ρ2(NE,r).
Let
(74)δ=∥S1+(E)-S2+(E)∥𝕃2(∂Br×∂Br).
Combining (17), (19), and (65), we get that
(75)|δhz0(λ)|≤δ∥∂ϕ1∂ν+-∂ϕ1∂ν-∥𝕃2(Br)∥∂ϕ2∂ν+-∂ϕ2∂ν-∥𝕃2(Br),(k,l)∈ΘE,|Imk|=|Iml|≠0,
where ϕj, j=1,2, denotes the solution of (18) with v=vj, satisfying
(76)ϕ1(x)=ψ~z0,1(x,-λ),ϕ2(x)=ψz0,2(x,λ),forx∈B¯r.
Using (21), (70a), and (70b) and the fact that C1(∂Br)⊂ℍ1(∂Br), we find that
(77)∥∂ϕj∂ν+-∂ϕj∂ν-∥𝕃2(Br)≤σc13(r)(1+E)exp(|λ|(4r2+4r)),λ∈ℂ,|λ|≥ρ1(NE,r,σ),j=1,2.
Here and bellow in this section the constant σ is the same that in (70a) and (70b).Combining (75), (77), we obtain that
(78)|δhz0(λ)|≤c14(E,r,σ)exp(|λ|(8r2+8r))δ,λ∈ℂ,|λ|≥ρ1(NE,r,σ).
Using (73) and (78), we get that
(79)|v2(z0)-v1(z0)|≤c14(E,r,σ)exp(|λ|(8r2+8r))δ+c12(NE,r)(ln(3|λ|))2|λ|3/4,z0∈Br,λ∈ℂ,|λ|≥ρ3(NE,r,σ)=max{ρ1,ρ2}.
We fix some τ∈(0,1) and let
(80)β=1-τ8r2+8r,λ=βln(3+δ-1),
where δ is so small that |λ|≥ρ3(NE,r,σ). Then due to (79), we have that
(81)∥v1-v2∥𝕃∞(Br)≤c14(E,r,σ)(3+δ-1)β(8r2+8r)δ+c12(NE,r)(ln(3βln(3+δ-1)))2(βln(3+δ-1))3/4=c14(E,r,σ)(1+3δ)1-τδτ+c12(NE,r)β-3/4(ln(3βln(3+δ-1)))2(ln(3+δ-1))3/4,
where τ, β, and δ are the same as in (80).Using (81), we obtain that
(82)∥v1-v2∥𝕃∞(Br)≤c15(N,E,r,σ)(ln(3+δ-1))-3/4×(ln(3ln(3+δ-1)))2,
for δ=∥S1+(E)-S2+(E)∥𝕃2(∂Br×∂Br)≤δ2(NE,r,σ), where δ2 is a sufficiently small positive constant. Estimate (82) in the general case (with modified c15) follows from (82) for δ≤δ2(NE,r,σ) and the property that ∥vj∥𝕃∞(Br)≤N.This completes the proof of (14).


## 8. Proof of Lemma7

In this section we assume for simplicity thatr=1 and therefore ∂Br=𝕊d-1.We fix an orthonormal basis in𝕃2(∂Br):
(83){fjp:j≥0;1≤p≤pj},fjpisasphericalharmonicofdegreej,
where pj is the dimension of the space of spherical harmonics of order j,
(84)pj=(j+d-1d-1)-(j+d-3d-1),
where
(85)(nk)=n(n-1)⋯(n-k+1)k!,forn≥0,(nk)=0,forn<0.
The precise choice of fjp is irrelevant for our purposes. Besides orthonormality, we only need fjp to be the restriction of a homogeneous harmonic polynomial of degree j to the sphere ∂Br and so |x|jfjp(x∣|x|) is harmonic pn ℝd. In the Sobolev spaces ℍs(∂Br) the norm is defined by
(86)∥∑j,p‍cjpfjp∥ℍs(∂Br)2=∑j,p‍(1+j)2s|cjp|2.
The solution ϕ of the exterior Dirichlet problem
(87)-Δϕ=Eϕ,inℝd∖B¯r,lim|x|→+∞|x|(d-1)/2(∂∂|x|ϕ-iEϕ)=0,ϕ|∂Br=u∈ℍ1(∂Br),
can be expressed in the following form (see, e.g., [1, 38]):
(88)ϕ=∑j,p‍cjpϕjp,
where cjp are expansion coefficients of u in the basis {fjp:j≥0;1≤p≤pj}, and
(89)ϕjpdenotesthesolutionof(89)withu=fjp,ϕjp(x)=hjp(|x|)fjp(x|x|),hjp(|x|)=|x|-(d-2)/2Hj+((d-2)/2)(1)(E|x|)Hj+((d-2)/2)(1)(E),
where Hμ(1) is the Hankel function of the first kind. Let
(90)ϕjp0(x)=|x|-j-d+2fjp(x|x|).
Note that ϕjp0 is harmonic in ℝd∖{0} and
(91)lim|x|→+∞|x|(d-1)/2(∂∂|x|ϕjp0-iEϕjp0)=0,forj+d-32>0.
Using the Green formula and the radiation condition for ϕjp,ϕjp0, we get that
(92)∫ℝd∖Br‍Eϕjpϕjp0dx=∫ℝd∖Br‍(Δϕjp0ϕjp-Δϕjpϕjp0)dx=∫∂Br‍(∂ϕjp0∂ν+ϕjp-∂ϕjp∂ν+ϕjp0)dx,forj+d-32>0.
Due to (89) and (90), we have that
(93)|∫∂Br‍∂ϕjp0∂ν+ϕjpdx|=(j+d-2)∫∂Br‍fjp2dx=j+d-2.
Using also the following property of the Hankel function of the first kind (see, e.g., [39]):
(94)|Hμ(1)(x)|isadecreasingfunctionofx,forx∈ℝ+,μ∈ℝ,
we get that
(95)|∫ℝd∖Br‍ϕjpϕjp0dx|=|∫1+∞‍t-j-d+2hjp(t)td-1dt|=|∫1+∞‍t-j-(d/2)Hj+((d-2)/2)(1)(Et)Hj+((d-2)/2)(1)(E)dt|≤∫1+∞‍t-j-(d/2)dt=1j+(d/2)-1≤2,forj+d-32>0.
Combining (89), (90), (92), (93), and (95), we obtain that
(96)|∫∂Br‍∂ϕjp0∂ν+ϕjpdx|=|hjp′(r)hjp(r)|≤j+d-2+2E,forj+d-32>0.Let consider the cases whenj+((d-3)/2)≤0.Case 1 (j=0,d=2).
Using the propertydH0(1)(t)/dt=-H1(1)(t), we get that
(97)hjp′(r)hjp(r)=EH1(1)(E)H0(1)(E).
We recall that functions H0(1) and H1(1) have the following asymptotic forms (see, e.g. [39]):
(98)H0(1)(t)~2iπln(t2),ast→+0,H0(1)(t)~2πtei(t-π/4),ast→+∞,H1(1)(t)~-iπ(2t),ast→+0,H1(1)(t)~2πtei(t-3π/4),ast→+∞.
Using (94) and (98), we get that for some c>0(99)H1(1)(t)H0(1)(t)≤c(1+1t).
Combining (97) and (99), we obtain that for j=0,d=2(100)|hjp′(r)hjp(r)|≤c(1+E).Case 2 (j=0,d=3).
We have that(101)Hj+((d-2)/2)(1)(t)=2πtei(t-π/2).
Using (89) and (101), we get that for j=0,d=3(102)hjp′(r)hjp(r)=-1+iE.
Combining (86)–(89), (96), (100), and (102), we get that for some constant c′=c′(d)>0:
(103)∥∂ϕ∂ν+|∂Br∥𝕃2(∂Br)2=∑j,p‍cjp2|hjp′(r)hjp(r)|2≤c′(1+E)2∑j,p‍(1+j)2cjp2.
Using (86) and (103), we obtain (21).



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