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

Abstract

We prove new global Hölder-logarithmic stability estimates for the near-field inverse scattering problem in dimension 𝑑≥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.

1. Introduction

We consider the Schrödinger equation: 𝐿𝜓=𝐸𝜓,𝐿=−Δ+𝑣(𝑥),𝑥∈ℝ^𝑑,𝑑≥2, (1) where 𝑣isreal-valued,𝑣∈𝕃^∞ℝ^𝑑,𝑣(𝑥)=𝑂|𝑥|^{−𝑑−𝜀},|𝑥|→∞,forsome𝜀>0. (2) We consider the resolvent 𝑅(𝐸) of the Schrödinger operator 𝐿 in 𝕃²(ℝ^𝑑): 𝑅(𝐸)=(𝐿−𝐸)⁻¹,𝐸∈ℂ⧵𝜎(𝐿), (3) where 𝜎(𝐿) is the spectrum of 𝐿 in 𝕃²(ℝ^𝑑). We assume that 𝑅(𝑥,𝑦,𝐸) denotes the Schwartz kernel of 𝑅(𝐸) as of an integral operator. We consider also 𝑅⁺𝑥,𝑦,𝐸=𝑅𝑥,𝑦,𝐸+𝑖0,𝑥,𝑦∈ℝ^𝑑,𝐸∈ℝ₊. (4) We recall that in the framework of (1) the function 𝑅⁺(𝑥,𝑦,𝐸) describes scattering of the spherical waves, 𝑅⁺₀𝑖𝑥,𝑦,𝐸=−4√𝐸2𝜋𝑥−𝑦^(𝑑−2)/2𝐻₍₎⁽¹⁾_𝑑−2/2√𝐸𝑥−𝑦, (5) generated by a source at 𝑦 (where 𝐻_𝜇⁽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 𝑆⁺𝑥,𝑦,𝐸=𝑅⁺𝑥,𝑦,𝐸−𝑅⁺₀𝑥,𝑦,𝐸,𝑥,𝑦∈𝜕𝐵_𝑟,𝐸∈ℝ₊,𝑟∈ℝ₊ (6) is considered as near-field scattering data for (1), where 𝐵_𝑟 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. Given 𝑆⁺ on 𝜕𝐵_𝑟×𝜕𝐵_𝑟 for some fixed 𝑟, 𝐸∈ℝ₊, find 𝑣 on 𝐵_𝑟.

This problem can be considered under the assumption that 𝑣 is a priori known on ℝ^𝑑⧵𝐵_𝑟. Actually, in the present paper we consider Problem 1 under the assumption that 𝑣≡0 on ℝ^𝑑⧵𝐵_𝑟 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 Problem 1 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 Problem 1 uniquely determine the Dirichlet-to-Neumann map in the case when 𝐸 is not a Dirichlet eigenvalue for operator 𝐿 in 𝐵_𝑟, 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 Problem 1 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.

Problem 1 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 energy 𝐸. 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 Problem 1 in dimension 𝑑=2, see Theorem 2.

2. Stability Estimates

We recall that if 𝑣 satisfies (2) and supp𝑣⊂𝐵_𝑟1 for some 𝑟₁>0, then 𝑆⁺(𝐸)isboundedin𝕃²𝜕𝐵_𝑟×𝜕𝐵_𝑟forany𝑟>𝑟₁, (7) 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 for 𝑑≥3

In this subsection we assume for simplicity that 𝑣∈𝕎^𝑚,1ℝ^𝑑forsome𝑚>𝑑,𝑣isreal-valued,supp𝑣⊂𝐵_𝑟1forsome𝑟₁>0, (8) where 𝕎^𝑚,1ℝ^𝑑=𝑣∶𝜕^𝐽𝑣∈𝕃¹ℝ^𝑑,|𝐽|≤𝑚,𝑚∈ℕ∪0, (9) where 𝐽∈(ℕ∪0)^𝑑,|𝐽|=^𝑑_𝑖=1𝐽_𝑖,𝜕^𝐽𝑣(𝑥)=𝜕^|𝐽|𝑣(𝑥)𝜕𝑥^𝐽1₁⋯𝜕𝑥^𝐽𝑑_𝑑. (10) Let ‖𝑣‖_𝑚,1=max_{|𝐽|≤𝑚}𝜕^𝐽𝑣_{𝕃1(ℝ𝑑)}. (11)

Note that (8)⇒(2).

Theorem 1. Let 𝐸>0 and 𝑟>𝑟₁ be given constants. Let dimension 𝑑≥3 and potentials 𝑣₁, 𝑣₂ satisfy (8). Let ‖𝑣_𝑗‖_𝑚,1≤𝑁,𝑗=1,2, for some 𝑁>0. Let 𝑆⁺₁(𝐸) and 𝑆⁺₂(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: 𝑣₂−𝑣_{1𝐿∞(𝐵𝑟)}≤𝐶₁(1+𝐸)^5/2𝛿^𝜏+𝐶₂(1+𝐸)^{(𝑠−𝑠∗)/2}ln3+𝛿^−1−𝑠, (12) where 𝑠^∗=(𝑚−𝑑)/𝑑, 𝛿=‖𝑆⁺₁(𝐸)−𝑆⁺₂(𝐸)‖_{𝕃2(𝜕𝐵𝑟×𝜕𝐵𝑟)}, and constants 𝐶₁,𝐶₂>0 depend only on 𝑁,𝑚,𝑑,𝑟,𝜏.

Proof of Theorem 1 is given in Section 5. This proof is based on results presented in Sections 3 and 4.

2.2. Estimates for 𝑑=2

In this subsection we assume for simplicity that 𝑣isreal-valued,𝑣∈𝐶²𝐵_𝑟1,supp𝑣⊂𝐵_𝑟1forsome𝑟₁>0. (13) Note also that (13)⇒(2).

Theorem 2. Let 𝐸>0 and 𝑟>𝑟₁ be given constants. Let dimension 𝑑=2 and potentials 𝑣₁, 𝑣₂ satisfy (13). Let ‖𝑣_𝑗‖_{𝐶2(𝐵𝑟)}≤𝑁,𝑗=1,2, for some 𝑁>0. Let 𝑆⁺₁(𝐸) and 𝑆⁺₂(𝐸) denote the near-field scattering data for 𝑣₁ and 𝑣₂, respectively. Then 𝑣₁−𝑣_{2𝐿∞(𝐵𝑟)}≤𝐶₃ln3+𝛿^−1−3/4ln3ln3+𝛿⁻¹², (14) where 𝛿=‖𝑆⁺₁(𝐸)−𝑆⁺₂(𝐸)‖_{𝕃2(𝜕𝐵𝑟×𝜕𝐵𝑟)} and constant 𝐶₃>0 depends only on 𝑁, 𝑚, 𝑟.

Proof of Theorem 2 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 Theorem 1 we obtain the following estimate (see formula (57)): 𝑣₁−𝑣_{2𝕃∞(𝐵𝑟)}≤𝐶₄(1+𝐸)²𝐸+𝜌²𝑒^{2𝜌(𝑟+1)}𝛿+𝐶₅𝐸+𝜌^{2−(𝑚−𝑑)/2𝑑}, (15) 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 Theorems 1 and 2 are fulfilled (in the cases of dimension 𝑑≥3 and 𝑑=2, resp.).

Consider the operators 𝑅_𝑗,𝑗=1,2, defined as follows 𝑅_𝑗𝜙(𝑥)=_𝜕𝐵𝑟𝑅⁺_𝑗𝑥,𝑦,𝐸𝜙𝑦𝑑𝑦,𝑥∈𝜕𝐵_𝑟,𝑗=1,2. (16) Note that 𝑅₁−𝑅_{2𝕃2(𝜕𝐵𝑟)}≤𝑆⁺₁(𝐸)−𝑆⁺₂(𝐸)_{𝕃2(𝜕𝐵𝑟)×𝕃2(𝜕𝐵𝑟)}. (17) We recall that (see [25]) for any functions 𝜙₁,𝜙₂∈𝐶(ℝ^𝑑), sufficiently regular in ℝ^𝑑⧵𝜕𝐵_𝑟 and satisfying −Δ𝜙+𝑣(𝑥)𝜙=𝐸𝜙,inℝ^𝑑⧵𝜕𝐵_𝑟,lim_{|𝑥|→+∞}|𝑥|^(𝑑−1)/2𝜕𝜕|𝑥|𝜙−𝑖√𝐸𝜙=0, (18) with 𝑣=𝑣₁ and 𝑣=𝑣₂, respectively, the following identity holds: _𝐵𝑟𝑣₂−𝑣₁𝜙₁𝜙₂=𝑑𝑥_𝜕𝐵𝑟𝜕𝜙₁𝜕𝜈₊−𝜕𝜙₁𝜕𝜈₋𝑅₁−𝑅₂𝜕𝜙₂𝜕𝜈₊−𝜕𝜙₂𝜕𝜈₋𝑑𝑥, (19) where where 𝜈₊ and 𝜈₋ are the outward and inward normals to 𝜕𝐵_𝑟, 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. Let 𝐸,𝑟>0 and 𝑑≥2. Then, there is a positive constant 𝐶₇ (depending only on 𝑟 and 𝑑) such that for any 𝜙∈𝐶(ℝ^𝑑⧵𝐵_𝑟) satisfying −Δ𝜙=𝐸𝜙,𝑖𝑛ℝ^𝑑⧵𝐵_𝑟,lim_{|𝑥|→+∞}|𝑥|^(𝑑−1)/2𝜕𝜕|𝑥|𝜙−𝑖√𝜙𝐸𝜙=0,_𝜕𝐵𝑟∈ℍ¹𝜕𝐵_𝑟, (20) the following inequality holds: 𝜕𝜙𝜕𝜈_{+𝜕𝐵𝑟𝕃2(𝜕𝐵𝑟)}≤𝐶₇(1+𝐸)𝜙_{𝜕𝐵𝑟ℍ1(𝜕𝐵𝑟)}, (21) where ℍ¹(𝜕𝐵_𝑟) denotes the standard Sobolev space on 𝜕𝐵_𝑟.

The proof of Lemma 7 is given in Section 8.

4. Faddeev Functions

In dimension 𝑑≥3, we consider the Faddeev functions ℎ, 𝜓, 𝐺 (see [6, 8, 30, 31]): ℎ(𝑘,𝑙)=(2𝜋)^−𝑑_ℝ𝑑𝑒^{−𝑖𝑙𝑥}𝑣(𝑥)𝜓(𝑥,𝑘)𝑑𝑥, (22) where 𝑘,𝑙∈ℂ^𝑑, 𝑘²=𝑙², Im𝑘=Im𝑙≠0, 𝜓(𝑥,𝑘)=𝑒^𝑖𝑘𝑥+_ℝ𝑑𝐺𝑣𝑦𝜓𝐺()𝑥−𝑦,𝑘𝑦,𝑘𝑑𝑦,𝑥,𝑘=𝑒^𝑖𝑘𝑥𝑔(),𝑔()()𝑥,𝑘𝑥,𝑘=−2𝜋^−𝑑_ℝ𝑑𝑒^𝑖𝜉𝑥𝑑𝜉𝜉²,+2𝑘𝜉 (23)(24) where 𝑥∈ℝ^𝑑, 𝑘∈ℂ^𝑑, Im𝑘≠0, 𝑑≥3,

One can consider (22), (23) assuming that 𝑣isasufficientlyregularfunctiononℝ^𝑑withsuffucientdecayatinfinity. (25) For example, in connection with Theorem 1, we consider (22), (23) assuming that 𝑣∈𝕃^∞𝐵_𝑟,𝑣≡0onℝ⧵𝐵_𝑟. (26)

We recall that (see [6, 8, 30, 31]) Δ+𝑘²𝐺(𝑥,𝑘)=𝛿(𝑥),𝑥∈ℝ^𝑑,𝑘∈ℂ^𝑑⧵ℝ^𝑑, (27) formula (23) at fixed 𝑘 is considered as an equation for 𝜓=𝑒^𝑖𝑘𝑥𝜇(𝑥,𝑘), (28) where 𝜇 is sought in 𝕃^∞(ℝ^𝑑); 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 Σ_𝐸=𝑘∈ℂ^𝑑∶𝑘²=𝑘²₁+⋯+𝑘²_𝑑,Θ=𝐸_𝐸=𝑘∈Σ_𝐸,𝑙∈Σ_𝐸,|𝑘|=||∶Im𝑘=Im𝑙Re𝑘²+|Im𝑘|^21/2. (29) Let 𝑣satisfy(8),‖𝑣‖_𝑚,1𝑣𝑝=()≤𝑁,2𝜋^−𝑑_ℝ𝑑𝑒^𝑖𝑝𝑥𝑣(𝑥)𝑑𝑥,𝑝∈ℝ^𝑑, (30) then we have that 𝜇(𝑥,𝑘)→1as|𝑘|→∞ (31) and, for any 𝜎>1, 𝜇(𝑥,𝑘)+∇𝜇(𝑥,𝑘)≤𝜎for|𝑘|≥𝜆₁(𝑁,𝑚,𝑑,𝑟,𝜎), (32) where 𝑥∈ℝ^𝑑, 𝑘∈Σ_𝐸; 𝑣𝑝=lim_{()𝑘,𝑙∈Θ𝐸||=||,𝑘−𝑙=𝑝Im𝑘Im𝑙→∞}ℎ(𝑘,𝑙),forany𝑝∈ℝ^𝑑,𝑣𝑝(≤𝑐−ℎ𝑘,𝑙)₁(𝑚,𝑑,𝑟)𝑁²𝐸+𝜌^21/2,for(𝑘,𝑙)∈Θ_𝐸||=||,𝑝=𝑘−𝑙,Im𝑘Im𝑙=𝜌,𝐸+𝜌²≥𝜆₂(),𝑝𝑁,𝑚,𝑑,𝑟₂≤4𝐸+𝜌². (33)(34)

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 Λ^−𝑠𝑔(𝑘)Λ^−𝑠_{𝕃2(ℝ𝑑)→𝕃2(ℝ𝑑)}=𝑂|𝑘|⁻¹as|𝑘|→∞,𝑘∈ℂ^𝑑⧵ℝ^𝑑, (35) 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 ℎ₂(𝑘,𝑙)−ℎ₁()=()𝑘,𝑙2𝜋^−𝑑_ℝ𝑑𝜓₁(𝑥,−𝑙)𝑣₂(𝑥)−𝑣₁(𝑥)𝜓₂()()𝑥,𝑘𝑑𝑥for𝑘,𝑙∈Θ_𝐸,|Im𝑘|=|Im𝑙|≠0,and𝑣₁,𝑣₂satisfying(25), (36) and, under assumptions of Theorem 1, 𝑣₁𝑝−𝑣₂𝑝−ℎ₁(𝑘,𝑙)+ℎ₂(≤𝑐𝑘,𝑙)₂(𝑚,𝑑,𝑟)𝑁𝑣₁−𝑣_{2𝕃∞(𝐵𝑟)}𝐸+𝜌^21/2for(𝑘,𝑙)∈Θ_𝐸,𝑝=𝑘−𝑙,|Im𝑘|=|Im𝑙|=𝜌,𝐸+𝜌²≥𝜆₃(𝑁,𝑚,𝑑,𝑟),𝑝²≤4𝐸+𝜌², (37) where ℎ_𝑗, 𝜓_𝑗 denote ℎ and 𝜓 of (22) and (23) for 𝑣=𝑣_𝑗,𝑗=1,2.

Formula (36) was given in [36]. Estimate (37) was given for example in [15].

5. Proof of Theorem 1

Let 𝕃^∞_𝜇ℝ^𝑑=𝑢∈𝕃^∞ℝ^𝑑∶‖𝑢‖_𝜇,‖𝑢‖<+∞_𝜇=esssup_{𝑝∈ℝ𝑑}1+𝑝^𝜇𝑢𝑝,𝜇>0. (38)

Note that 𝑤∈𝕎^𝑚,1ℝ^𝑑⇒𝑤∈𝕃^∞_𝜇ℝ^𝑑∩𝐶ℝ^𝑑,‖𝑤‖_𝜇≤𝑐₃(𝑚,𝑑)‖𝑤‖_𝑚,1,for𝜇=𝑚, (39) where 𝕎^𝑚,1, 𝕃^∞_𝜇 are the spaces of (9), (38), 𝑤𝑝=(2𝜋)^−𝑑_ℝ𝑑𝑒^𝑖𝑝𝑥𝑤(𝑥)𝑑𝑥,𝑝∈ℝ^𝑑. (40)

Using the inverse Fourier transform formula 𝑤(𝑥)=_ℝ𝑑𝑒^{−𝑖𝑝𝑥}𝑤𝑝𝑑𝑝,𝑥∈ℝ^𝑑, (41) we have that 𝑣₁−𝑣_{2𝕃∞(𝐵𝑟)}≤sup_{𝑥∈𝐵𝑟ℝ𝑑}𝑒^{−𝑖𝑝𝑥}𝑣₂𝑝−𝑣₁𝑝𝑑𝑝≤𝐼₁(𝜅)+𝐼₂(𝜅),forany𝜅>0, (42) where 𝐼₁(𝜅)=_{|𝑝|≤𝜅}𝑣₂𝑝−𝑣₁𝑝𝐼𝑑𝑝,₂(𝜅)=_{|𝑝|≥𝜅}𝑣₂𝑝−𝑣₁𝑝𝑑𝑝. (43)

Using (39), we obtain that 𝑣₂𝑝−𝑣₁𝑝≤2𝑐₃(𝑚,𝑑)𝑁1+𝑝^−𝑚,𝑝∈ℝ^𝑑. (44) Let 𝑐₄=_{𝑝∈ℝ𝑑,|𝑝|=1}𝑑𝑝. (45) Combining (43), (44), we find that, for any 𝜅>0, 𝐼₂(𝜅)≤2𝑐₃(𝑚,𝑑)𝑁𝑐_4𝜅^+∞𝑑𝑡𝑡^{𝑚−𝑑+1}≤2𝑐₃(𝑚,𝑑)𝑁𝑐₄1𝑚−𝑑𝜅^𝑚−𝑑. (46)

Due to (37), we have that 𝑣₂𝑝−𝑣₁𝑝≤ℎ₂(𝑘,𝑙)−ℎ₁(+𝑐𝑘,𝑙)₂(𝑚,𝑑,𝑟)𝑁𝑣₁−𝑣_{2𝕃∞(𝐵𝑟)}𝐸+𝜌^21/2,for(𝑘,𝑙)∈Θ_𝐸,𝑝=𝑘−𝑙,|Im𝑘|=|Im𝑙|=𝜌,𝐸+𝜌²≥𝜆₃(𝑁,𝑚,𝑑,𝑟),𝑝²≤4𝐸+𝜌². (47)

Let 𝛿=𝑆⁺₁(𝐸)−𝑆⁺₂(𝐸)_{𝕃2(𝜕𝐵𝑟×𝜕𝐵𝑟)}. (48) Combining (17), (19), and (36), we get that ℎ₂(𝑘,𝑙)−ℎ₁(𝑘,𝑙)≤𝛿𝜕𝜙₁𝜕𝜈₊−𝜕𝜙₁𝜕𝜈_{−𝕃2(𝐵𝑟)}𝜕𝜙₂𝜕𝜈₊−𝜕𝜙₂𝜕𝜈_{−𝕃2(𝐵𝑟)},(𝑘,𝑙)∈Θ_𝐸,|Im𝑘|=|Im𝑙|≠0, (49) where 𝜙_𝑗, 𝑗=1,2, denotes the solution of (18) with 𝑣=𝑣_𝑗, satisfying 𝜙_𝑗(𝑥)=𝜓_𝑗(𝑥,𝑘),for𝑥∈𝐵_𝑟. (50) Using (21), (32), and the fact that 𝐶¹(𝜕𝐵_𝑟)⊂ℍ¹(𝜕𝐵_𝑟), we find that 𝜕𝜙_𝑗𝜕𝜈₊−𝜕𝜙_𝑗𝜕𝜈_{−𝕃2(𝐵𝑟)}≤𝜎𝑐₅(𝑟,𝑑)(1+𝐸)exp(|Im𝑘|(𝑟+1)),𝑘∈Σ_𝐸,|𝑘|≥𝜆₁(𝑁,𝑚,𝑑,𝑟,𝜎),𝑗=1,2. (51) Here and bellow in this section the constant 𝜎 is the same that in (32).

Combining (49) and (51), we obtain that ℎ₂(𝑘,𝑙)−ℎ₁(𝑘,𝑙)≤𝑐²₅𝜎²(1+𝐸)²𝑒^{2𝜌(𝑟+1)}()𝛿,for𝑘,𝑙∈Θ_𝐸,𝜌=|Im𝑘|=|Im𝑙|,𝐸+𝜌²≥𝜆²₁(𝑁,𝑚,𝑑,𝑟,𝜎). (52) Using (47), (52), we get that 𝑣₂𝑝−𝑣₁𝑝≤𝑐²₅𝜎²(1+𝐸)²𝑒^{2𝜌(𝑟+1)}𝛿+𝑐₂(𝑚,𝑑,𝑟)𝑁𝑣₁−𝑣_{2𝕃∞(𝐵1)}𝐸+𝜌^21/2,𝑝∈ℝ^𝑑,𝑝²≤4𝐸+𝜌²,𝐸+𝜌²≥max𝜆²₁,𝜆₃. (53) Let 1𝜀=2𝑐₂(𝑚,𝑑,𝑟)𝑁𝑐₆^1/𝑑,𝑐₆=_{𝑝∈ℝ𝑑,|𝑝|≤1}𝑑𝑝, (54) and 𝜆₄(𝑁,𝑚,𝑑,𝑟,𝜎)>0 be such that 𝐸+𝜌²≥𝜆₄()⇒𝑁,𝑚,𝑑,𝑟,𝜎𝐸+𝜌²≥𝜆²₁(𝑁,𝑚,𝑑,𝑟,𝜎),𝐸+𝜌²≥𝜆₃(),𝜀𝑁,𝑚,𝑑,𝑟𝐸+𝜌^21/2𝑑2≤4𝐸+𝜌². (55) Using (43), (53), we get that 𝐼₁(𝜅)≤𝑐₆𝜅^𝑑𝑐²₅𝜎²(1+𝐸)²𝑒^{2𝜌(𝑟+1)}𝛿+𝑐₂(𝑚,𝑑,𝑟)𝑁𝑣₁−𝑣_{2𝕃∞(𝐵1)}𝐸+𝜌^21/2,𝜅>0,𝜅²≤4𝐸+𝜌²,𝐸+𝜌²≥𝜆₄(𝑁,𝑚,𝑑,𝑟,𝜎). (56) Combining (42), (46), and (56) for 𝜅=𝜀(𝐸+𝜌²)^1/2𝑑 and (55), we get that 𝑣₁−𝑣_{2𝕃∞(𝐵𝑟)}≤𝑐₇(𝑁,𝑚,𝑑,𝑟,𝜎)(1+𝐸)²𝐸+𝜌²𝑒^{2𝜌(𝑟+1)}𝛿+𝑐₈(𝑁,𝑚,𝑑)𝐸+𝜌^{2−(𝑚−𝑑)/2𝑑}+12𝑣₁−𝑣_{2𝕃∞(𝐵𝑟)},𝐸+𝜌²≥𝜆₄(𝑁,𝑚,𝑑,𝑟,𝜎). (57)

Let 𝜏^′∈(0,1), 𝛽=1−𝜏^′2(𝑟+1),𝜌=𝛽ln3+𝛿⁻¹, (58) and 𝛿₁=𝛿₁(𝑁,𝑚,𝑑,𝜎,𝑟,𝜏^′)>0 be such that 𝛿∈0,𝛿₁⇒𝐸+𝛽ln3+𝛿⁻¹²≥𝜆₄(𝑁,𝑚,𝑑,𝑟,𝜎),𝐸+𝛽ln3+𝛿⁻¹²≤(1+𝐸)𝛽ln3+𝛿⁻¹². (59) Then for the case when 𝛿∈(0,𝛿₁), due to (57), we have that 12𝑣₁−𝑣_{2𝕃∞(𝐵𝑟)}≤𝑐₇(1+𝐸)²𝐸+𝛽ln3+𝛿^−121/23+𝛿^{−12𝛽(𝑟+1)}𝛿+𝑐₈𝐸+𝛽ln3+𝛿^{−12−(𝑚−𝑑)/2𝑑}=𝑐₇(1+𝐸)²𝐸+𝛽ln3+𝛿^−121/2(1+3𝛿)^1−𝜏′𝛿^𝜏′+𝑐₈𝐸+𝛽ln3+𝛿^{−12−(𝑚−𝑑)/2𝑑}. (60) Combining (59) and (60), we obtain that for 𝑠∈[0,𝑠^∗],𝜏∈(0,𝜏^′) and 𝛿∈(0,𝛿₁) the following estimate holds: 𝑣₂−𝑣_{1𝐿∞(𝐵𝑟)}≤𝑐₉(1+𝐸)^5/2𝛿^𝜏+𝑐₁₀(1+𝐸)^{(𝑠−𝑠∗)/2}ln3+𝛿^−1−𝑠, (61) 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 𝑣_{𝑗𝕃∞(𝐵𝑟)}≤𝑐₁₁(𝑚,𝑑)𝑁. (62)

This completes the proof of (12).

6. Buckhgeim-Type Analogs of the Faddeev Functions

Let us identify ℝ² with ℂ and use coordinates 𝑧=𝑥₁+𝑖𝑥₂, 𝑧=𝑥₁−𝑖𝑥₂, where (𝑥₁,𝑥₂)∈ℝ². Following [21–23, 37], we consider the functions 𝐺_𝑧0, 𝜓_𝑧0, 𝜓_𝑧0, 𝛿ℎ_𝑧0 going back to Buckhgeim’s paper [3] and being analogs of the Faddeev functions: 𝜓_𝑧0(𝑧,𝜆)=𝑒^{𝜆(𝑧−𝑧0)2}+_𝐵𝑟𝐺_𝑧0(𝑧,𝜁,𝜆)𝑣(𝜁)𝜓_𝑧0()𝜓𝜁,𝜆𝑑Re𝜁𝑑Im𝜁,_𝑧0(𝑧,𝜆)=𝑒^{𝜆(𝑧−𝑧0)2}+_𝐵𝑟𝐺_𝑧0(𝑧,𝜁,𝜆)𝑣(𝜁)𝜓_𝑧0()𝐺𝜁,𝜆𝑑Re𝜁𝑑Im𝜁,_𝑧0()=1𝑧,𝜁,𝜆4𝜋²_𝐵𝑟𝑒^{−𝜆(𝜂−𝑧0)2+𝜆(𝜂−𝑧0)2}𝑑Re𝜂𝑑Im𝜂𝑧−𝜂𝜂−𝜁×𝑒^{𝜆(𝑧−𝑧0)2−𝜆(𝜁−𝑧0)2},𝑧=𝑥₁+𝑖𝑥₂,𝑧₀∈𝐵_𝑟,𝜆∈ℂ, (63)(64) where 𝑣 satisfies (13), 𝛿ℎ_𝑧0(𝜆)=_𝐵𝑟𝜓_𝑧0,1(𝑧,−𝜆)𝑣₂(𝑧)−𝑣₁(𝑧)×𝜓_𝑧0,2(𝑧,𝜆)𝑑Re𝑧𝑑Im𝑧,𝜆∈ℂ, (65) where 𝑣₁, 𝑣₂ satisfy (13) and 𝜓_𝑧0,1, 𝜓_𝑧0,2 denote 𝜓_𝑧0, 𝜓_𝑧0 of (63) for 𝑣=𝑣₁ and 𝑣=𝑣₂, respectively.

We recall that (see [21, 22]): (i) the function 𝐺_𝑧0 satisfies the equations 4𝜕²𝜕𝑧𝜕𝑧𝐺_𝑧0()(),4𝜕𝑧,𝜁,𝜆=𝛿𝑧−𝜁₂𝜕𝜁𝜕𝜁𝐺_𝑧0(𝑧,𝜁,𝜆)=𝛿(𝑧−𝜁), (66)where 𝑧,𝑧₀,𝜁∈𝐵_𝑟, 𝜆∈ℂ, and 𝛿 is the Dirac delta function;(ii) formulas (63) at fixed 𝑧₀ and 𝜆 are considered as equations for 𝜓_𝑧0, 𝜓_𝑧0 in 𝐿^∞(𝐵_𝑟);(iii) as a corollary of (63), (64), (66), the functions 𝜓_𝑧0, 𝜓_𝑧0 satisfy (1) in 𝐵_𝑟 for 𝐸=0 and 𝑑=2;(iv) the function 𝛿ℎ_𝑧0 is similar to the right side of (36).

Let potentials 𝑣,𝑣₁,𝑣₂∈𝐶²(𝐵_𝑟), and ‖𝑣‖_𝐶²_(𝐵𝑟)≤𝑁,𝑣_{𝑗𝐶2(𝐵𝑟)}𝑣≤𝑁,𝑗=1,2,₁−𝑣_{2𝜕𝐵𝑟}𝜕=0,𝑣𝜕𝜈¹−𝑣_{2𝜕𝐵𝑟}=0, (67) then we have that 𝜓_𝑧0(𝑧,𝜆)=𝑒^{𝜆(𝑧−𝑧0)2}𝜇_𝑧0(),𝜓𝑧,𝜆_𝑧0(𝑧,𝜆)=𝑒^{𝜆(𝑧−𝑧0)2}𝜇_𝑧0(),𝜇𝑧,𝜆_𝑧0(𝑧,𝜆)→1,𝜇_𝑧0(𝑧,𝜆)→1as|𝜆|→∞ (68)(69) and, for any 𝜎>1, 𝜇_𝑧0(𝑧,𝜆)+∇𝜇_𝑧0(𝜇𝑧,𝜆)≤𝜎,_𝑧0(𝑧,𝜆)+∇𝜇_𝑧0(𝑧,𝜆)≤𝜎, (70a)(70b)where ∇=(𝜕/𝜕𝑥₁,𝜕/𝜕𝑥₂), 𝑧=𝑥₁+𝑖𝑥₂, 𝑧₀∈𝐵_𝑟, 𝜆∈ℂ, |𝜆|≥𝜌₁(𝑁,𝑟,𝜎); 𝑣₂𝑧₀−𝑣₁𝑧₀=lim_𝜆→∞2𝜋|𝜆|𝛿ℎ_𝑧0(𝜆)forany𝑧₀∈𝐵_𝑟,𝑣₂𝑧₀−𝑣₁𝑧₀−2𝜋|𝜆|𝛿ℎ_𝑧0(𝜆≤𝑐₁₂)(𝑁,𝑟)(ln(3|𝜆|))²|𝜆|^3/4,for𝑧₀∈𝐵_𝑟,|𝜆|≥𝜌₂(𝑁,𝑟). (71)(72) Formulas (68) can be considered as definitions of 𝜇_𝑧0, 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 Theorem 2

We suppose that  𝜓_𝑧0,1(⋅,−𝜆),𝜓_𝑧0,2(⋅,𝜆),𝛿ℎ_𝑧0(𝜆) are defined as in Section 6 but with 𝑣_𝑗−𝐸 in place of 𝑣_𝑗, 𝑗=1,2. Note that functions  𝜓_𝑧0,1(⋅,−𝜆),𝜓_𝑧0,2(⋅,𝜆) satisfy (1) in 𝐵_𝑟 with 𝑣=𝑣_𝑗, 𝑗=1,2, respectively. We also use the notation 𝑁_𝐸=𝑁+𝐸. Then, using (72), we have that 𝑣₂𝑧₀−𝑣₁𝑧₀−2𝜋|𝜆|𝛿ℎ_𝑧0(𝜆≤𝑐₁₂)𝑁_𝐸,𝑟(ln(3|𝜆|))²|𝜆|^3/4,for𝑧₀∈𝐵_𝑟,|𝜆|≥𝜌₂𝑁_𝐸,𝑟. (73) Let 𝛿=𝑆⁺₁(𝐸)−𝑆⁺₂(𝐸)_{𝕃2(𝜕𝐵𝑟×𝜕𝐵𝑟)}. (74) Combining (17), (19), and (65), we get that 𝛿ℎ_𝑧0(𝜆)≤𝛿𝜕𝜙₁𝜕𝜈₊−𝜕𝜙₁𝜕𝜈_{−𝕃2(𝐵𝑟)}𝜕𝜙₂𝜕𝜈₊−𝜕𝜙₂𝜕𝜈_{−𝕃2(𝐵𝑟)},(𝑘,𝑙)∈Θ_𝐸,|Im𝑘|=|Im𝑙|≠0, (75) where 𝜙_𝑗, 𝑗=1,2, denotes the solution of (18) with 𝑣=𝑣_𝑗, satisfying 𝜙₁(𝑥)=𝜓_𝑧0,1(),𝜙𝑥,−𝜆₂(𝑥)=𝜓_𝑧0,2(𝑥,𝜆),for𝑥∈𝐵_𝑟. (76) Using (21), (70a), and (70b) and the fact that 𝐶¹(𝜕𝐵_𝑟)⊂ℍ¹(𝜕𝐵_𝑟), we find that 𝜕𝜙_𝑗𝜕𝜈₊−𝜕𝜙_𝑗𝜕𝜈_{−𝕃2(𝐵𝑟)}≤𝜎𝑐₁₃(𝑟)(1+𝐸)exp|𝜆|4𝑟²,|𝜆|+4𝑟𝜆∈ℂ,≥𝜌₁𝑁_𝐸,𝑟,𝜎,𝑗=1,2. (77) Here and bellow in this section the constant 𝜎 is the same that in (70a) and (70b).

Combining (75), (77), we obtain that 𝛿ℎ_𝑧0(𝜆)≤𝑐₁₄(𝐸,𝑟,𝜎)exp|𝜆|8𝑟²|𝜆|+8𝑟𝛿,𝜆∈ℂ,≥𝜌₁𝑁_𝐸,𝑟,𝜎. (78) Using (73) and (78), we get that 𝑣₂𝑧₀−𝑣₁𝑧₀≤𝑐₁₄(𝐸,𝑟,𝜎)exp|𝜆|8𝑟²𝛿+𝑐+8𝑟₁₂𝑁_𝐸,𝑟(ln(3|𝜆|))²|𝜆|^3/4,𝑧₀∈𝐵_𝑟,𝜆∈ℂ,|𝜆|≥𝜌₃𝑁_𝐸,𝑟,𝜎=max𝜌₁,𝜌₂. (79) We fix some 𝜏∈(0,1) and let 𝛽=1−𝜏8𝑟²+8𝑟,𝜆=𝛽ln3+𝛿⁻¹, (80) where 𝛿 is so small that |𝜆|≥𝜌₃(𝑁_𝐸,𝑟,𝜎). Then due to (79), we have that 𝑣₁−𝑣_{2𝕃∞(𝐵𝑟)}≤𝑐₁₄(𝐸,𝑟,𝜎)3+𝛿^{−1𝛽(8𝑟2+8𝑟)}𝛿+𝑐₁₂𝑁_𝐸,𝑟ln3𝛽ln3+𝛿⁻¹²𝛽ln3+𝛿^−13/4=𝑐₁₄(𝐸,𝑟,𝜎)(1+3𝛿)^1−𝜏𝛿^𝜏+𝑐₁₂𝑁_𝐸,𝑟𝛽^−3/4ln3𝛽ln3+𝛿⁻¹²ln3+𝛿^−13/4, (81) where 𝜏, 𝛽, and 𝛿 are the same as in (80).

Using (81), we obtain that 𝑣₁−𝑣_{2𝕃∞(𝐵𝑟)}≤𝑐₁₅(𝑁,𝐸,𝑟,𝜎)ln3+𝛿^−1−3/4×ln3ln3+𝛿⁻¹², (82) for 𝛿=‖𝑆⁺₁(𝐸)−𝑆⁺₂(𝐸)‖_{𝕃2(𝜕𝐵𝑟×𝜕𝐵𝑟)}≤𝛿₂(𝑁_𝐸,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(𝑁_𝐸,𝑟,𝜎) and the property that ‖𝑣_𝑗‖_{𝕃∞(𝐵𝑟)}≤𝑁.

This completes the proof of (14).

8. Proof of Lemma 7

In this section we assume for simplicity that 𝑟=1 and therefore 𝜕𝐵_𝑟=𝕊^𝑑−1.

We fix an orthonormal basis in 𝕃²(𝜕𝐵_𝑟): 𝑓_𝑗𝑝∶𝑗≥0;1≤𝑝≤𝑝_𝑗,𝑓_𝑗𝑝isasphericalharmonicofdegree𝑗, (83) where 𝑝_𝑗 is the dimension of the space of spherical harmonics of order 𝑗, 𝑝_𝑗=−𝑗+𝑑−1𝑑−1𝑗+𝑑−3𝑑−1, (84) where 𝑛𝑘=𝑛(𝑛−1)⋯(𝑛−𝑘+1)𝑛𝑘𝑘!,for𝑛≥0,=0,for𝑛<0. (85) The precise choice of 𝑓_𝑗𝑝 is irrelevant for our purposes. Besides orthonormality, we only need 𝑓_𝑗𝑝 to be the restriction of a homogeneous harmonic polynomial of degree 𝑗 to the sphere 𝜕𝐵_𝑟 and so |𝑥|^𝑗𝑓_𝑗𝑝(𝑥∣|𝑥|) is harmonic pn ℝ^𝑑. In the Sobolev spaces ℍ^𝑠(𝜕𝐵_𝑟) the norm is defined by _𝑗,𝑝𝑐_𝑗𝑝𝑓_𝑗𝑝²_{ℍ𝑠𝜕𝐵𝑟}=_𝑗,𝑝1+𝑗^2𝑠𝑐_𝑗𝑝². (86) The solution 𝜙 of the exterior Dirichlet problem −Δ𝜙=𝐸𝜙,inℝ^𝑑⧵𝐵_𝑟,lim_{|𝑥|→+∞}|𝑥|^(𝑑−1)/2𝜕𝜕|𝑥|𝜙−𝑖√𝜙𝐸𝜙=0,_𝜕𝐵𝑟=𝑢∈ℍ¹𝜕𝐵_𝑟, (87) can be expressed in the following form (see, e.g., [1, 38]): 𝜙=_𝑗,𝑝𝑐_𝑗𝑝𝜙_𝑗𝑝, (88) where 𝑐_𝑗𝑝 are expansion coefficients of 𝑢 in the basis {𝑓_𝑗𝑝∶𝑗≥0;1≤𝑝≤𝑝_𝑗}, and 𝜙_𝑗𝑝denotesthesolutionof(89)with𝑢=𝑓_𝑗𝑝,𝜙_𝑗𝑝(𝑥)=ℎ_𝑗𝑝(|𝑥|)𝑓_𝑗𝑝𝑥|𝑥|,ℎ_𝑗𝑝(|𝑥|)=|𝑥|^{−(𝑑−2)/2}𝐻₎₎⁽¹⁾_{𝑗+((𝑑−2/2}√𝐸|𝑥|𝐻₎₎⁽¹⁾_{𝑗+((𝑑−2/2}√𝐸, (89) where 𝐻_𝜇⁽¹⁾ is the Hankel function of the first kind. Let 𝜙⁰_𝑗𝑝(𝑥)=|𝑥|^{−𝑗−𝑑+2}𝑓_𝑗𝑝𝑥|𝑥|. (90) Note that 𝜙⁰_𝑗𝑝 is harmonic in ℝ^𝑑⧵{0} and lim_{|𝑥|→+∞}|𝑥|^(𝑑−1)/2𝜕𝜕|𝑥|𝜙⁰_𝑗𝑝−𝑖√𝐸𝜙⁰_𝑗𝑝=0,for𝑗+𝑑−32>0. (91) Using the Green formula and the radiation condition for 𝜙_𝑗𝑝,𝜙⁰_𝑗𝑝, we get that _{ℝ𝑑⧵𝐵𝑟}𝐸𝜙_𝑗𝑝𝜙⁰_𝑗𝑝=𝑑𝑥_{ℝ𝑑⧵𝐵𝑟}Δ𝜙⁰_𝑗𝑝𝜙_𝑗𝑝−Δ𝜙_𝑗𝑝𝜙⁰_𝑗𝑝=𝑑𝑥_𝜕𝐵𝑟𝜕𝜙⁰_𝑗𝑝𝜕𝜈₊𝜙_𝑗𝑝−𝜕𝜙_𝑗𝑝𝜕𝜈₊𝜙⁰_𝑗𝑝𝑑𝑥,for𝑗+𝑑−32>0. (92) Due to (89) and (90), we have that _𝜕𝐵𝑟𝜕𝜙⁰_𝑗𝑝𝜕𝜈₊𝜙_𝑗𝑝𝑑𝑥=𝑗+𝑑−2_𝜕𝐵𝑟𝑓²_𝑗𝑝𝑑𝑥=𝑗+𝑑−2. (93) Using also the following property of the Hankel function of the first kind (see, e.g., [39]): 𝐻_𝜇⁽¹⁾(𝑥)isadecreasingfunctionof𝑥,for𝑥∈ℝ₊,𝜇∈ℝ, (94) we get that _{ℝ𝑑⧵𝐵𝑟}𝜙_𝑗𝑝𝜙⁰_𝑗𝑝𝑑𝑥=₁^+∞𝑡^{−𝑗−𝑑+2}ℎ_𝑗𝑝(𝑡)𝑡^𝑑−1=𝑑𝑡_1+∞𝑡^{−𝑗−(𝑑/2)}𝐻₎₎⁽¹⁾_{𝑗+((𝑑−2/2}√𝐸𝑡𝐻₎₎⁽¹⁾_{𝑗+((𝑑−2/2}√𝐸≤𝑑𝑡_1+∞𝑡^{−𝑗−(𝑑/2)}1𝑑𝑡=𝑗+(𝑑/2)−1≤2,for𝑗+𝑑−32>0. (95) Combining (89), (90), (92), (93), and (95), we obtain that _𝜕𝐵𝑟𝜕𝜙⁰_𝑗𝑝𝜕𝜈₊𝜙_𝑗𝑝=ℎ𝑑𝑥^′𝑗𝑝(𝑟)ℎ_𝑗𝑝(𝑟)≤𝑗+𝑑−2+2𝐸,for𝑗+𝑑−32>0. (96)

Let consider the cases when 𝑗+((𝑑−3)/2)≤0.

Case 1 (𝑗=0,𝑑=2). Using the property 𝑑𝐻₀⁽¹⁾(𝑡)/𝑑𝑡=−𝐻₁⁽¹⁾(𝑡), we get that ℎ^′_𝑗𝑝(𝑟)ℎ_𝑗𝑝(𝑟)=√𝐸𝐻₁⁽¹⁾√𝐸𝐻₀⁽¹⁾√𝐸. (97) We recall that functions 𝐻₀⁽¹⁾ and 𝐻₁⁽¹⁾ have the following asymptotic forms (see, e.g. [39]): 𝐻₀⁽¹⁾(𝑡)∼2𝑖𝜋𝑡ln2𝐻,as𝑡→+0,₀₍₁₎(𝑡)∼2𝑒𝜋𝑡^{𝑖(𝑡−𝜋/4)}𝐻,as𝑡→+∞,₁₍₁₎(𝑡)𝑖∼−𝜋2𝑡𝐻,as𝑡→+0,₁₍₁₎(𝑡)∼2𝑒𝜋𝑡^{𝑖(𝑡−3𝜋/4)},as𝑡→+∞. (98) Using (94) and (98), we get that for some 𝑐>0𝐻₁⁽¹⁾(𝑡)𝐻₀⁽¹⁾(𝑡)1≤𝑐1+𝑡. (99) Combining (97) and (99), we obtain that for 𝑗=0,𝑑=2ℎ^′_𝑗𝑝(𝑟)ℎ_𝑗𝑝(𝑟)≤𝑐1+√𝐸. (100)

Case 2 (𝑗=0,𝑑=3). We have that 𝐻₎₎⁽¹⁾_{𝑗+((𝑑−2/2}(𝑡)=2𝑒𝜋𝑡^{𝑖(𝑡−𝜋/2)}. (101) Using (89) and (101), we get that for 𝑗=0,𝑑=3ℎ^′_𝑗𝑝(𝑟)ℎ_𝑗𝑝(𝑟)=−1+𝑖√𝐸. (102) Combining (86)–(89), (96), (100), and (102), we get that for some constant 𝑐^′=𝑐^′(𝑑)>0: 𝜕𝜙𝜕𝜈_{+𝜕𝐵𝑟}²_{𝕃2𝜕𝐵𝑟}=_𝑗,𝑝𝑐²_𝑗𝑝ℎ^′_𝑗𝑝(𝑟)ℎ_𝑗𝑝(𝑟)²≤𝑐^′(1+𝐸)²_𝑗,𝑝1+𝑗²𝑐²_𝑗𝑝. (103) Using (86) and (103), we obtain (21).

Acknowledgment

This work was partially supported by FCP Kadry no. 14.A18.21.0866.