The ∂ -theory for Inverse Problems Associated with Schr¨ odinger Operators

The ∂ -theory for Inverse Problems Associated with Schr¨ odinger Operators

on Hyperbolic Spaces

By

HiroshiIsozaki^∗

Abstract

An analogue of the Faddeev scattering amplitude is introduced for Schr¨odinger operators on hyperbolic spaces. It satisfies a ∂-equation and enables us to derive an integral representation of the potential.

§1. Introduction

In the present paper, we are concerned with the inverse problem associated with Schr¨odinger operators on hyperbolic spaces. The most fundamental object in scattering theory is the S-matrix. For the Schr¨odinger operator inRⁿ, it is a unitary operator on L²(Sⁿ⁻¹) having the following expression

(1.1) S(E)f(θ) =f(θ) +C_E

Sⁿ⁻¹

A(E; θ, ω)f(ω)dω, f ∈L²(Sⁿ⁻¹), whereC_Eis a constant depending only on the energyE >0 and the scattering amplitudeA(E; θ, ω) is observed from the asymptotic behavior of the solution to the Schr¨odinger equation

(1.2) (−∆ +V(x))ϕ=Eϕ

in the following manner :

(1.3) ϕ(x;E, ω)∼e^i√Eω·x+C_E e^i√Er

r^(n−1)/2A(E; θ, ω)

Communicated by T. Kawai. Received February 24, 2005. Revised February 13, 2006, April 13, 2006.

2000 Mathematics Subject Classification(s): Primary 35R30; Secondary 81U40.

∗Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan.

e-mail: isozakih@math.tsukuba.ac.jp

asr=|x| → ∞, θ=x/r. Thisϕis obtained by solving the Lippman-Schwinger equation :

(1.4) ϕ(x) =e^i√Eω·x−

Rⁿ

G₀(x−y, E)V(y)ϕ(y)dy, where G₀(x, E) is the Green function for−∆−E defined by (1.5) G₀(x, E) = (2π)⁻ⁿ

Rⁿ

e^ix·ξ ξ²−E−i0dξ.

Here and in the sequel forζ= (ζ₁,· · ·, ζ_n)∈Cⁿ, we denoteζ²=_n

i=1ζ_i². The inverse problem for the Schr¨odinger operator aims at constructing V(x) from the S-matrix. When n = 1, the well-known theory of Gel’fand- Levitan-Marchenko provides us with the necessary and sufficient condition for a functionS(E) to be the S-matrix of a Schr¨odinger operator and an algorithm for the reconstruction ofV(x).

The multi-dimensional inverse problem has not been solved yet completely as in the 1-dimensional case. The main difficulty arises from the overdetermi- nacy ; the scattering amplitude ˜A(E;θ, ω) is a function of 2n−1 parameters while the potential V(x) depends on n variables. Therefore for a function f(E, θ, ω) on (0,∞)×Sⁿ⁻¹×Sⁿ⁻¹ to be the scattering amplitude associated with a Schr¨odinger operator, f must satisfy a sort of compatibility condition, which is still unknown. However, there is a series of deep results related to inverse problems in multi-dimensions, the main idea of which consists in using exponentially growing solutions for the Schr¨odinger equation (1.2). In inverse boundary value problems in a bounded domain, it is called the complex geo- metrical optics solution (see [SyUh]). In the inverse scattering problem, it is commonly called the∂-theory ([Na1], [Na2], [KhNo]), although the pioneering work of Faddeev [Fa] does not use this term.

In the ∂-approach of inverse scattering, instead of A(E; θ, ω), one uses Faddeev’s scattering amplitude :

(1.6) A(ξ, ζ) =

Rⁿ

e^{−ix·(ξ+ζ)}V(x)ψ(x, ζ)dx, ξ∈Rⁿ, ζ∈Cⁿ where ζ²=E, andψ(x, ζ) is a solution to the equation

(1.7) ψ(x, ζ) =e^ix·ζ−

Rⁿ

G(x−y, ζ)V(y)ψ(y, ζ)dy, G(x, ζ) being Faddeev’s Green function defined by

(1.8) G(x, ζ) = (2π)⁻ⁿ

Rⁿ

e^ix·(ξ+ζ) ξ²+ 2ζ·ξdξ.

This functionA(ξ, ζ) has the following features :

(i) It is natural to regard A(ξ, ζ) as a function on the fiber bundle M=

∪ξ{ξ} × Vξ, whereξ varies over the base spaceRⁿ and the fiberVξ is defined by

(1.9) V_ξ ={ζ∈Cⁿ;ζ²=E, ξ²+ 2ζ·ξ= 0,Imζ= 0}. As a 1-form on M, it satisfies a∂-equation

(1.10)

∂_ζA(ξ, ζ) =−(2π)¹⁻ⁿ n j=1

Rⁿ

A(ξ−η, ζ+η)A(η, ζ)η_jδ(η²+ 2ζ·η)dη

dζ_j. (ii) When n ≥ 3, the Fourier transform of the potential V is recovered from A(ξ, ζ) in the following way :

(1.11) Vˆ(ξ) = (2π)^−n/2 lim

|ζ|→∞, ζ∈Vξ

A(ξ, ζ).

Consequently, by virtue of a generalization of Bochner-Martinelli’s formula onV_ξ, we have an integral representation ofV(x) in terms ofA(ξ, ζ).

(iii) The ∂-equation characterizes the Faddeev scattering amplitude.

Namely, the equation (1.10) is a necessary and sufficient condition for a func- tion A(ξ, ζ) on the fiber bundleM to be the scattering amplitude associated with a Schr¨odinger operator onRⁿ.

These ideas have been found and confirmed in various levels. For the details see [NaAb], [BeCo], [Na1], [No], [Gr] and especially the introduction of [KhNo]. See also [Ha], [We].

The purpose of the present paper is to generalize (a part of) these results for the Schr¨odinger operator on the hyperbolic space Hⁿ. In our previous works [Is1], [Is2], we have seen a close connection between the inverse problem on the Euclidean space and that on the hyperbolic space. Namely, the inverse boundary value problem inRⁿis equivalent to the one inHⁿ, or the hyperbolic quotient manifolds under the action of discrete group of translations, which then turns out to be equivalent to the inverse scattering problem. Whenn≥3, this latter can be solved by passing to the Faddeev scattering amplitude for the Floquet operators. However, the inversion procedures are not quite constructive and essentially the uniqueness has been proven.

In this paper we show that the Green function for the gauge-transformed Laplacian on Hⁿ satisfies a ∂-equation (Theorem 2.7). We then introduce an analogue of Faddeev scattering amplitude and derive a ∂-formula for it

(Theorem 3.4). When n = 3, this leads to an integral representation of the potential in terms of the Faddeev scattering amplitude (Theorems 3.5 and 3.7).

The counter part ofA(ξ, ζ) introduced in this paper is a triple{B_II, B_IJ, B_JI} ((3.26)–(3.28)) living on a simple line bundle and one can make use of the standard generalized Cauchy formula onCto derive the integral representation of the potential. We allow the potentialV to be complex-valued.

Two interesting problems remain open. One is the relation between the physical scattering amplitude and the Faddeev scattering amplitude. When the potential is compactly supported, these two scattering amplitudes determine each other through the Dirichlet-Neumann map for the boundary value problem on a bounded domain which contains the support of the potential. However, a direct link between them in the case of potentials of long-tail is still unknown.

The other problem is the characterization of Faddeev scattering amplitude in terms of the ∂-equation. We shall return to these problems elsewhere.

We mainly work in H³, although many preliminary results are proven in general dimensions. The reason of the restriction to n= 3 is that the decay estimate for the Green operator (Theorems 2.8) is proved only whenn≤3.

In §2, we prepare basic estimates for the Green operator of the gauge- transformed Laplacian on Hⁿ and derive the ∂-equation. In §3 we introduce the Faddeev scattering amplitude and derive its∂-equation and integral repre- sentation formulas of the potential. In§4, we show that the Faddeev scattering amplitude and the Dirichlet-Neumann map of the boundary value problem on a bounded domain determine each other.

§2. Green Operators

§2.1. Modified Bessel functions

Let J_ν(y) be the Bessel function of order ν. For y > 0 modified Bessel functions are defined by

I_ν(y) =e^−νπi/2J_ν(iy), ν∈C, (2.1)

K_ν(y) = π 2

I_−ν(y)−I_ν(y)

sin(νπ) , ν ∈Z, (2.2)

K_n(z) =K_−n(z) = lim

ν→nK_ν(z), n∈Z.

(2.3)

They are linearly independent solutions of the equation (2.4) y²u+yu−(y²+ν²)u= 0.

They are analytic in the complex plane with cut along the negative real axis and

(2.5) I_ν(z) =

z 2

_ν^∞

k=0

(z²/4)^k k! Γ(ν+k+ 1). If−π <argz < π, we have

I_ν(e^mπiz) =e^νmπiI_ν(z), K_ν(e^mπiz) =e^−νmπiK_ν(z)−πisin(νmπ)

sin(νπ) I_ν(z).

(See [Wa], p. 80). In particular, forr >0,

I_ν(ir) =e^νπiI_ν(−ir) =e^νπi/2J_ν(r), (2.6)

K_ν(ir) =e^−νπiK_ν(−ir)−πiI_ν(−ir).

(2.7)

The following asymptotic expansions are well-known (see [Wa], p. 202) : I_ν(z)∼ e^z

√2πz +e−z+(ν+1/2)πi

√2πz , |z| → ∞, −π

2 <argz < 3π 2 , (2.8)

I_ν(z)∼ e^z

√2πz +e−z−(ν+1/2)πi

√2πz , |z| → ∞, −3π

2 <argz < π 2, (2.9)

K_ν(z)∼ π

2ze^−z, |z| → ∞, −π <argz < π.

(2.10)

The formula (2.5) implies

(2.11) I_ν(z)∼ 1

Γ(ν+ 1) z

2 _ν

, z→0, and forν ∈Z,

(2.12) K_ν(z)∼ π 2 sin(νπ)

1 Γ(1−ν)

z 2

_−ν

− 1

Γ(1 +ν) z

2 _ν

, z→0.

Whenn= 0,1,2,· · ·,K_n(z) has the following expression (see [Le], p. 110) : K_n(z) = (−1)ⁿ⁻¹I_n(z) logz

2 +1

2

n−1

k=0

(−1)^k(n−k−1)!

k!

z 2

_2k−n

−(−1)ⁿ⁻¹ 2

∞ k=0

ψ(k+ 1) +ψ(k+n+ 1) k!(k+n)!

z 2

_2k+n , (2.13)

ψ(1) =−γ, ψ(k+ 1) =−γ+ 1 + 1

2+· · ·+1

k, k= 1,2,· · ·, (2.14)

γ being Euler’s constant. This implies that whenz→0

(2.15) K_n(z)∼

−logz, n= 0, 2ⁿ⁻¹(n−1)!z⁻ⁿ, n= 1,2,· · ·.

As was discussed in [Is1] and [Is2], one can solve the inverse boundary value problem in Rⁿ by imbedding it into Hⁿ, where we encounter the case ν = 1/2, and the above functions are written in terms of elementary functions :

I_1/2(z) = 2 πzsinhz, (2.16)

K_1/2(z) = π 2ze^−z, (2.17)

J_1/2(z) = 2 πzsinz.

(2.18)

§2.2. 1-dimensional operator Let

(2.19) E= (n−1)²

4 −ν²,

and for a complex parameterζ= 0 satisfying Reζ≥0, consider the differential operator

(2.20) L₀(ζ) =y²(−∂_y²+ζ²) + (n−2)y∂_y−E on (0,∞), wheren≥2 is an integer and∂_y=∂/∂y. By (2.4)

y^(n−1)/2I_ν(ζy), y^(n−1)/2K_ν(ζy) are linearly independent solutions ofL₀(ζ)u= 0.

The above constantEcorresponds to the energy parameter for the Laplace- Beltrami operator H₀ on Hⁿ, whose spectrum is the interval [(n−1)²/4,∞).

Accordingly, we consider two cases :

(2.21) ν ∈iR\ {0}, or ν ∈(0,∞)\Z.

In the former caseE belongs to the spectrum of H₀, and in the latter case to the resolvent set. In this paper we do not deal with the case ν∈Zin order to

avoid the logarithmic singularity ofK_n(z). We define the Green kernel of the 1-dimensional operator (2.20) by

(2.22) G₀(y, y;ζ) =

(yy)^(n−1)/2K_ν(ζy)I_ν(ζy), y > y>0, (yy)^(n−1)/2I_ν(ζy)K_ν(ζy), y> y >0, and introduce the Green operator

(2.23) G₀(ζ)f(y) = _∞

0

G₀(y, y;ζ)f(y) dy (y)ⁿ. Since

y^(n−1)/2I_ν(ζy) y^(n−1)/2K_ν(ζy)

−

y^(n−1)/2I_ν(ζy)

y^(n−1)/2K_ν(ζy)

=−yⁿ⁻², we have for f ∈C₀^∞((0,∞))

(2.24) L₀(ζ)G₀(ζ)f =f.

Throughout the paper we assume that ν satisfies (2.21), although we do not mention it specifically.

Lemma 2.1. The Green function G₀(y, y;ζ) is analytic in ζ when Reζ >0. There exists a constantC=C_ν >0 such that the inequalities

|G₀(y, y;ζ)| ≤C(yy)^(n−1)/2, (2.25)

|G₀(y, y;ζ)| ≤ C

|ζ|(yy)^(n−2)/2, (2.26)

∂

∂ζG₀(y, y;ζ)≤ C

|ζ|(yy)^(n−2)/2(y+y) (2.27)

hold fory, y>0 andζ such that Reζ≥0.

Proof. The analyticity is obvious. By virtue of (2.8)–(2.15), we have

|I_ν(z)| ≤C |z|

1 +|z| _Reν

(1 +|z|)^−1/2e^Rez, (2.28)

|K_ν(z)| ≤C |z|

1 +|z| _−Reν

(1 +|z|)^−1/2e^−Rez. (2.29)

Sincet/(1 +t) is monotone increasing fort≥0, by (2.28), (2.29) and Reν≥0 we have if y > y >0,

|K_ν(ζy)I_ν(ζy)| ≤C e^{−Reζ(y−y)}

(1 +|ζy|)^1/2(1 +|ζy|)^1/2. It then follows that

(2.30) |G₀(y, y;ζ)| ≤C(yy)^(n−1)/2 e^{−Reζ|y−y|}

(1 +|ζy|)^1/2(1 +|ζy|)^1/2, which proves (2.25) and (2.26). Using

2I_ν(z) =I_ν−1(z) +I_ν+1(z), (2.31)

−2K_ν(z) =K_ν−1(z) +K_ν+1(z), (2.32)

(see [Le], p. 110) and (2.28), (2.29) as well as (2.15), we have

|zI_ν(z)| ≤C |z|

1 +|z| _Reν

(1 +|z|)^1/2e^Rez, (2.33)

|zK_ν(z)| ≤C |z|

1 +|z| _−Reν

(1 +|z|)^1/2e^−Rez, (2.34)

which imply ∂

∂ζI_ν(ζy)≤ C

|ζ|

|ζy| 1 +|ζy|

_Reν

(1 +|ζy|)^1/2e^Reζy, (2.35)

∂

∂ζK_ν(ζy)≤ C

|ζ|

|ζy| 1 +|ζy|

_−Reν

(1 +|ζy|)^1/2e^−Reζy. (2.36)

This together with (2.28), (2.29) and an elementary inequality 1 +|ζy|

1 +|ζy| _1/2

≤

1 + y y

_1/2

≤ y+y (yy)^1/2 proves (2.27).

§2.3. Green operator on Hⁿ

Let us construct a Green operator of

(2.37) H₀(θ) =y²(−∂_y²+ (−i∂_x+θ)²) + (n−2)y∂_y.

Forθ, θ∈Cⁿ⁻¹, we put θ·θ =

n−1

i=1

θ_iθ_i, θ²=θ·θ, and define for ξ∈Rⁿ⁻¹

(2.38) ζ(ξ, θ) =

(ξ+θ)², where we take the branch of √

· such that Re√

· ≥ 0, i.e. √ z =√

re^iϕ/2 for z=re^iϕ,−π < ϕ < π. We define

G₀(θ)f(x, y) = (2π)^−(n−1)/2

Rⁿ⁻¹

e^ix·ξ

G₀(ζ(ξ, θ)) ˆf(ξ,·)

(y)dξ, (2.39)

fˆ(ξ, y) = (2π)^−(n−1)/2

Rⁿ⁻¹

e^−ix·ξf(x, y)dx.

(2.40)

By (2.24) and (2.25), we have

(2.41) (H₀(θ)−E)G₀(θ)f =f, ∀f ∈C₀^∞(Rⁿ₊).

Let us remark that when θ ∈Rⁿ⁻¹ and ν =iσ with σ > 0 (or σ < 0), G₀(θ) is the incoming (or outgoing) Green operator ofH₀(θ)−E :

G₀(θ) = (H₀(θ)−(E∓i0))⁻¹,

where the right-hand side exists on a certain Banach space (see [Is1]), which we now explain.

Fors∈R, we introduce the following function space : (2.42) f ∈L^2,s ⇐⇒

Rⁿ₊

(1 +|logy|)^2s|f(x, y)|²dxdy yⁿ <∞

equipped with the obvious norm. In the following, for two Banach spaces X andY,B(X;Y) denotes the totality of bounded operators fromX toY.

Lemma 2.2. Let s >1/2. Then there exists a constantC =C_s,ν >0 such that

G₀(θ)_B(L^2,s_,L^2,−s₎≤C, ∀θ∈Cⁿ⁻¹. Proof. Letu=G₀(θ)f. Since

ˆ u(ξ, y) =

_∞

0

G₀(y, y;ζ(ξ, θ)) ˆf(ξ, y) dy (y)ⁿ,

we have by using (2.25),

|u(ξ, y)ˆ |²≤Cyⁿ⁻¹ _∞

0

(1 +|logy|)^2s|fˆ(ξ, y)|² dy (y)ⁿ. Integration with respect toξandy proves the lemma.

This Green operatorG₀(θ) has an integral kernel (2.43) G₀(x, y, x, y;θ) = (2π)⁻⁽ⁿ⁻¹⁾

Rⁿ⁻¹

e^i(x−x)·ξG₀(y, y;ζ(ξ, θ))dξ.

It has the following estimates.

Lemma 2.3. (1)For anyθ∈Cⁿ⁻¹, there exists a constantC=C_θ>0 independent of x, x, y, y such that

|G₀(x, y, x, y;θ)| ≤C(yy)^(n−1)/2

|y−y|ⁿ⁻¹ e^C|y−y|.

(2) Fix y, y >0 andθ∈Cⁿ⁻¹ arbitrarily. Then there exist h₁(x), h₂(x) such that

G₀(x, x, y, y;θ) =h₁(x−x) +h₂(x−x),

where h₁(x)∈H^m(Rⁿ⁻¹), ∀m >0,h₂(x)∈C^∞(Rⁿ⁻¹\ {0})and satisfies

|∂_x^αh₂(x)| ≤C_α|x|^−N, ∀α, N > n−2.

Proof. There exists a constant C >0 such that Reζ(ξ, θ)≥ |ξ|/2−C.

This together with the estimate |G₀(y, y;ζ)| ≤ C(yy)^(n−1)/2e^{−Reζ|y−y|} (see (2.30)) proves the assertion (1).

Take χ₀, χ_∞ ∈ C^∞(Rⁿ⁻¹) such that χ₀(ξ) +χ_∞(ξ) = 1 on Rⁿ⁻¹ and χ₀(ξ) = 1 for|ξ|< C, χ₀(ξ) = 0 for|ξ|>2C, whereC is chosen large enough so thatζ(ξ, θ) is smooth on|ξ|> C. SplitG₀(y, y;ζ(ξ, θ)) into two parts :

g₀(ξ) =χ₀(ξ)G₀(y, y;ζ(ξ, θ)), g_∞(ξ) =χ_∞(ξ)G₀(y, y;ζ(ξ, θ)).

We then have

G₀(x, x, y, y;θ) = ˆg₀(x−x) + ˆg_∞(x−x), and ˆg₀∈H^m(Rⁿ⁻¹),∀m >0, by (2.25). Let ˆξ=ξ/|ξ|. Using

ζ(ξ, θ) =|ξ|+ ˆξ·θ+O(1/|ξ|), as |ξ| → ∞,

we have the asymptotic expansions I_ν(ζ(ξ, θ)y) = e^{(|ξ|+ˆξ·θ)y}

2π|ξ|y

1 + a₁( ˆξ)

|ξ| +a₂( ˆξ)

|ξ|² +· · · , K_ν(ζ(ξ, θ)y) = π

2|ξ|ye^{−(|ξ|+ˆξ·θ)y}

1 + b₁( ˆξ)

|ξ| +b₂( ˆξ)

|ξ|² +· · · ,

a_m( ˆξ), b_m( ˆξ) being smooth functions. Hence we have the asymptotic expansion g_∞(ξ)∼χ_∞(ξ)e^{−|ξ||y−y|}

∞ m=1

c_m( ˆξ)

|ξ|^m , where c_m( ˆξ) is a smooth function. By induction one can show

|∂_ξ^αe^{−|ξ||y−y|}| ≤C_α|ξ|^−|α|. Hence we have

∂^α_ξ

χ_∞(ξ)e^{−|ξ||y−y|}c_m( ˆξ)

|ξ|^m ≤C_α(1 +|ξ|)^−m−|α|. We now usee^ix·ξ=−|x|⁻²∆_ξe^ix·ξ and integrate by parts to see that

e^ix·ξχ_∞(ξ)e^{−|ξ||y−y|}c_m( ˆξ)

|ξ|^m dξ≤C_N|x|^−N, which proves the assertion (2).

§2.4. ∂-equation

Forθ=θ_R+iθ_I ∈Cⁿ⁻¹, let∂_θ be defined as follows : (2.44) ∂_θ=

∂

∂θ₁,· · ·, ∂

∂θ_n−1

, ∂

∂θ_j = 1 2

∂

∂θ_Rj +i ∂

∂θ_Ij

.

We are going to compute∂_θG₀(θ). Note that iff(z) is analytic,f(ζ(ξ, θ)) has singularities on the set{θ∈Cⁿ⁻¹; (ξ+θ)²≤0}.

Lemma 2.4. Let f(z) be an analytic function on {z∈C; Rez > 0} satisfying

sup

|z|<r|f(z)|<∞, ∀r >0.

Forθ=θ_R+iθ_I ∈Cⁿ⁻¹ such that θ_I = 0we put r_θ(ξ) =

|θ_I|²− |ξ+θ_R|², (2.45)

M_θ=

ξ∈Rⁿ⁻¹;θ_I ·(ξ+θ_R) = 0,|ξ+θ_R|<|θ_I| , (2.46)

and define a compactly supported distributionT_θ(ξ)by (2.47) T_θ(ξ), ϕ(ξ)=

Mθ

ϕ(ξ)i(ξ+θ)

2|θ_I| dM_θ(ξ), ∀ϕ∈C₀^∞(Rⁿ⁻¹), dM_θ(ξ) being the measure on M_θ induced from the Lebesgue measure dξ on Rⁿ⁻¹. Then regarding f(ζ(ξ, θ))as a distribution with respect to ξ ∈ Rⁿ⁻¹ depending on a parameter θ∈Cⁿ⁻¹, we have forθ_I = 0

(2.48) ∂_θf(ζ(ξ, θ)) = [f(ir_θ(ξ))−f(−ir_θ(ξ))]T_θ(ξ).

Proof. Takeχ(t)∈C^∞(R) such thatχ(t) = 1 (|t|>2),χ(t) = 0 (|t|<1) and letχ (t) =χ(t/). Sinceζ(ξ, θ) is analytic with respect toθifθ_I·(ξ+θ_R)= 0, we have

∂_θχ (θ_I·(ξ+θ_R))f(ζ(ξ, θ)) = i 2χ

θ_I·(ξ+θ_R)

(ξ+θ)f(ζ(ξ, θ)).

We put

τ=|θ_I|, α=θ_I/|θ_I|, p^⊥=p−(p·α)α, p∈Rⁿ⁻¹.

Let k₁=α·(ξ+θ_R), k₂= (ξ+θ_R)^⊥. Again letting k₁ =η₁ with >0, we have

ζ(ξ, θ)²=²η₁²+k₂²−τ²+ 2iτ η₁. Therefore when→0

ζ(ξ, θ)→







k₂²−τ² if |k₂| ≥τ,

sgn(η₁)i

τ²−k₂² if |k₂|< τ, where sgn(η₁) = 1 ifη₁>0, sgn(η₁) =−1 ifη₁<0. Let

g(k) =f(ζ(ξ, θ)), ψ(k) =ϕ(ξ).

Then we have

∂_θ

χ (θ_I·(ξ+θ_R))f(ζ(ξ, θ))ϕ(ξ)dξ

= i 2

χ(τ η₁)

η₁α+k₂−iθ_I

g(η₁, k₂)ψ(η₁, k₂)dη₁dk₂.

Let us recall here that the boundary value on the imaginary axis f(is) = lim _→0f(is +) exists almost every where (see e.g. [Hof], p. 38). Since _∞

−∞χ(τ η₁)dη₁= 0, we have

|k2|>τ

χ(τ η₁)

η₁α+k₂−iθ_I

g(η₁, k₂)ψ(η₁, k₂)dη₁dk₂

→

|k2|>τ

χ(τ η₁)

k₂−iθ_I f

k₂²−τ²

ψ(0, k₂)dη₁dk₂= 0.

On the other hand, since _∞

0 χ(τ η₁)dη₁ = 1/τ, ₀

−∞χ(τ η₁)dη₁ =−1/τ, the integral over the region{|k₂|< τ}converges to

η1>0,|k2|<τ

χ(τ η₁)i(k₂−iθ_I)

2 f

i

τ²−k²₂

ψ(0, k₂)dη₁dk₂ +

η1<0,|k2|<τ

χ(τ η₁)i(k₂−iθ_I)

2 f

−i

τ²−k²₂

ψ(0, k₂)dη₁dk₂

= 1 τ

|k2|<τ

f

i

τ²−k²₂

−f −i

τ²−k₂²

i(k₂−iθ_I)

2 ψ(0, k₂)dk₂. From this the lemma follows immediately.

We put

(2.49) D_θ(ξ) = [G₀(ir_θ(ξ))−G₀(−ir_θ(ξ)]i(ξ+θ) 2|θ_I| . By virtue of (2.22), (2.39) and (2.48), we have formally (2.50) ∂_θG₀(θ)f(x, y) = (2π)^−(n−1)/2

Mθ

e^ix·ξ

D_θ(ξ) ˆf(ξ,·)

(y)dM_θ(ξ).

Let us give a precise meaning to this operator. For t, s∈R, we introduce the function spaces :

H^(±)_t,s f ⇐⇒

Rⁿ₊

(1 +|x|)^t(1 +|logy|)^s(1 +y) y^1/2

_±2

|f(x, y)|²dxdy yⁿ <∞, (2.51)

H_s^(±)=H_s,s^(±) (2.52)

equipped with the obvious norm. We define the operatorχ (θ_I·(D+θ_R)) by (2.53) (χ (θ_I·(D+θ_R))ϕ) (x)

= (2π)^−(n−1)/2

Rⁿ⁻¹

e^ix·ξχ (θ_I·(ξ+θ_R)) ˆϕ(ξ)dξ,

where χ is the function given in the proof of Lemma 2.4, and put (2.54) G_0, (θ) =χ (θ_I·(D+θ_R))G₀(θ).

The following theorem gives the procedure for defining∂_θG₀(θ).

Theorem 2.5. Lets >1/2and supposef∈H⁽⁺⁾s . We putu =G_0, (θ)f, u=G₀(θ)f and

v(x, y;θ) = (2π)^−(n−1)/2

Mθ

e^ix·ξ

D_θ(ξ) ˆf(ξ,·)

(y)dM_θ(ξ).

Then

(1) u →uin L^2,−s.

(2) u is strongly differentiable inH⁽⁻⁾_0,s with respect toθif θ_I = 0.

(3) ∂_θu →v inH⁽⁻⁾s .

Proof. The assertion (1) is obvious. Let us prove (2). If >0 is fixed andθ varies over a bounded set, we have on suppχ (θ_I·(ξ+θ_R))

1/|ζ(ξ, θ)| ≤C (1 +|ξ|)⁻¹.

Letd_θ=∂/∂θor∂/∂θ. Then by virtue of (2.26), (2.27) and the above estimate, we have

|d_θuˆ (ξ, y;θ)| ≤C _∞

0

(yy)^(n−2)/2(1 +y)(1 +y)|fˆ(ξ, y)| dy (y)ⁿ, which implies for s >1/2

y (1 +y)²

|d_θuˆ (ξ, y;θ)|²

yⁿ ≤ C

y _∞

0

(1 +|logy|)^2s(1 +y)²

y |fˆ(ξ, y)|² dy (y)ⁿ. Therefore by Lebesgue’s theorem, u(x, y;θ) is aH⁽⁻⁾_0,s-valuedC¹-function ofθ ifθ_I = 0, and we have

(2.55) ∂_θu (x, y;θ) = (2π)^−(n−1)/2

Rⁿ₊

e^ix·ξ i 2χ

θ_I·(ξ+θ_R)

(ξ+θ)

×G₀(y, y;ζ(ξ, θ)) ˆf(ξ, y)dξdy (y)ⁿ.

Let us show the assertion (3). Letθvary over the ball{|θ|< C₀}and pick ψ₀, ψ_∞ ∈C^∞(Rⁿ) such that ψ₀(ξ) = 1 if|ξ|< 5C₀, ψ₀(ξ) = 0 if |ξ| >6C₀, andψ_∞(ξ) = 1−ψ₀(ξ). We put

ˆ

v⁽⁰⁾(ξ, y;θ) =ψ₀(ξ)∂_θuˆ (ξ, y;θ), (2.56)

ˆ

v^(∞)(ξ, y;θ) =ψ_∞(ξ)∂_θuˆ (ξ, y;θ).

(2.57)

We first show that v^(∞) → 0 in H⁽⁻⁾s as → 0. Assume for the notational simplicity that θ_I/|θ_I| = α = (1,0,· · ·,0), and let k₁ = α·(ξ+θ_R), k₂ = (ξ+θ_R)^⊥. We put

(2.58) w^(∞)(x₁, k₂, y;θ) = (2π)^−1/2 _∞

−∞

e^ix1k1vˆ^(∞)(k₁, k₂, y;θ)dk₁. By the change of variablek₁=η₁, we then have lettingτ=|θ_I|

w^(∞)(x₁, k₂, y;θ) = (2π)^−1/2i 2

R²₊

e^{i η1x1}χ(τ η₁)(η₁α+k₂−iθ_I)

×ψ_∞(η₁, k₂)G₀(y, y;ζ(η₁, k₂, θ)) ˆf(η₁, k₂, y)dη₁dy (y)ⁿ . (2.59)

On suppψ_∞(η₁, k₂), we have

1/|ζ(η₁, k₂, θ))| ≤C(1 +|k₂|)⁻¹, where Cis independent of >0. We also note that letting

f˜(x₁, k₂, y) = (2π)^−(n−2)/2

Rⁿ⁻²

e^−ix2·k2f(x₁, x₂, y)dx₂, we have for s >1/2

(2.60) sup

k1

|fˆ(k₁, k₂, y)| ≤C _∞

−∞x₁^2s|f(x˜ ₁, k₂, y)|²dx_{1 1/2}

.

Here and in the following we writex= (1 +|x|²)^1/2. Therefore by (2.26) the integrand of the right-hand side of (2.59) is dominated from above by

C|χ(τ η₁)|(yy)^(n−2)/2|fˆ(η₁, k₂, y)|

(y)ⁿ ≤Cy^(n−2)/2|χ(τ η₁)|(1 +|logy|)^−s

√y

× _∞

−∞x₁^2s(1 +|logy|)^2s|f(x˜ ₁, k₂, y)|² (y)ⁿ⁺¹ dx₁

_1/2 .

We then see by Schwarz’ inequality that the right-hand side is integrable with respect toη₁ andy for a.e. k₂. Therefore by Lebesgue’s convergence theorem

and the argument in the proof of Lemma 2.4, w^(∞) →0 pointwise as → 0.

We also have

|w^(∞)(x₁, k₂, y;θ)|

≤Cy^(n−2)/2

R²₊

(1 +|logy|)^2sx₁^2s

y |f˜(x₁, k₂, y)|²dx₁dy (y)ⁿ

_1/2 . Again by virtue of Lebesgue’s theorem we have

Rⁿ₊x₁^−2s(1 +|logy|)^−2s|w^(∞)(x₁, k₂, y;θ)|²

yⁿ⁻¹ dx₁dk₂dy→0, which provesv^(∞)→0 inH⁽⁻⁾s .

We finally consider v⁽⁰⁾(ξ, y;θ). Let w⁽⁰⁾(x₁, k₂, y;θ)

= (2π)^−1/2 _∞

−∞

e^ix1k1ˆv⁽⁰⁾(k₁, k₂, y;θ)dk₁

= (2π)^−1/2i 2

R²₊

e^{i η1x1}χ(τ η₁)(η₁α+k₂−iθ_I)ψ₀(η₁, k₂)

× G₀(y, y;ζ(η₁, k₂, θ)) ˆf(η₁, k₂, y)dη₁dy (y)ⁿ .

Then by the same argument as in the proof of Lemma 2.4 we have if|k₂|> τ, w⁽⁰⁾(x₁, k₂, y;θ)→0

pointwise as →0, and if|k₂|< τ, w⁽⁰⁾(x₁, k₂, y;θ)→(2π)^−1/2

τ

_∞

0

G₀

y, y;i

τ²−k²₂

−G₀

y, y;−i

τ²−k₂²

i(k₂−iθ_I) 2

fˆ(0, k₂, y) dy (y)ⁿ pointwise as →0. Moreover by (2.25) and (2.60), we have

|w⁽⁰⁾(x₁, k₂, y;θ)|

≤Cy^(n−1)/2×

R²₊

(1 +|logy|)^2sx₁^2s|f˜(x₁, k₂, y)|²dx₁dy (y)ⁿ

_1/2 . By Lebesgue’s convergence theorem, v⁽⁰⁾ →vin Hs⁽⁻⁾.

Let us rewrite∂_θG₀(θ) more explicitly.

Lemma 2.6. Forr >0

G₀(ir)−G₀(−ir)

f(y) =−πi _∞

0

(yy)^(n−1)/2J_ν(ry)J_ν(ry)f(y) dy (y)ⁿ. Proof. In view of (2.6) and (2.7), we have

K_ν(iry)I_ν(iry)−K_ν(−iry)I_ν(−iry)

=−e^νπiπiI_ν(−iry)I_ν(−iry)

=−πiJ_ν(ry)J_ν(ry).

The lemma then follows from (2.22).

The above lemma and (2.49) together with Theorem 2.5 then imply the following

Theorem 2.7. Let f ∈ Hs⁽⁺⁾with s >1/2 and suppose θ_I = 0. Then : (1) For >0,G_0, (θ)f is an H⁽⁻⁾_0,s-valued C¹-function of θ.

(2) G_0, (θ)f →G₀(θ)f inL^2,−s as→0.

(3) When → 0, ∂_θG_0, (θ)f converges in Hs⁽⁻⁾. Denoting this limit by

∂_θG₀(θ)f, we have the following formula:

∂_θG₀(θ)f=− πi

(2π)^(n−1)/2 _Mθ×(0,∞)e^ix·k(yy)^(n−1)/2

·J_ν(r_θ(k)y)J_ν(r_θ(k)y) ˆf(k, y)i k+θ 2|θ_I|

dM_θ(k)dy (y)ⁿ . Let us remark that in Theorem 2.1 of [IsUh], the analyticity with respect to z∈C of the Green operatorG₀(zα), α∈Sⁿ⁻¹ is stated. It is not correct and the above Theorem 2.7 gives the correct assertion.

§2.5. Perturbed Green operator

From now on we restrict the space dimension to 2 or 3. For n = 3 and s, t≥0, we introduce the following function spaces :

(2.61) W_t,s^(±)u⇐⇒

R³₊

(1 +|x|)^2t(1 +|logy|)^2s y

_±1

|u(x, y)|²dxdy y³ <∞ equipped with the obvious norm. Forn= 2 ands≥0, we define

(2.62) W_s^(±)u⇐⇒

R²₊

(1 +|logy|)^2s y

_±1

|u(x, y)|²dxdy y² <∞.