The diffraction of light is considered for a plane screen with an open bounded aperture. The corresponding solution behind the screen is given explicitly in terms of the Fourier transforms of the tangential components of the electric boundary field in the aperture. All components of the electric as well as the magnetic field vector are considered. We introduce solutions with global finite energy behind the screen and describe them in terms of two boundary potential functions. This new approach leads to a decoupling of the vectorial boundary equations in the aperture in the case of global finite energy. For the physically admissible solutions, that is, the solutions with local finite energy, we derive a characterisation in terms of the electric boundary fields.

1. Introduction

This paper deals with the classical diffraction problem for electromagnetic waves passing a bounded aperture in an ideally conducting plane screen. We treat the problem within the exact theory, that is, we consider the corresponding solutions of the time harmonic Maxwell equations that fulfil the correct boundary conditions on the screen.

The problem of diffraction of electromagnetic waves by an infinite slit has been treated by the Fourier method in the papers [1, 2]. In [1] especially representations of the solutions that fulfil a certain energy condition have been given in terms of distributional electric boundary fields satisfying special regularity properties. In [2] mapping properties of the corresponding boundary operators between Sobolev spaces have been studied. These Sobolev spaces have been chosen such that the corresponding diffraction solutions satisfy the correct physical energy condition.

While the slit problem treated in [1, 2] can be decoupled into two scalar problems by considering two kinds of polarisations of the electromagnetic field, in the case of a bounded aperture such a decoupling is not possible in general. However, for the latter case we derive a new kind of decoupling of the vectorial system which can be performed if and only if the condition of global finite energy in part (b) of Definition 2.5 is fulfilled, see Theorem 3.2.

Finally we study the condition of local finite energy which covers all physically admissible solutions. Here we give a characterisation of solutions with local finite energy in terms of a regularity property of the electric boundary fields, see Theorem 4.1. This is done in a self-contained way by using the Paley-Wiener theorem for distributions defined on the bounded aperture as well as a special contour integration method in the Fourier domain.

2. Electromagnetic Diffraction by an Aperture in a Plane Screen

We start with an informal physical description of the electromagnetic diffraction problem and fix some notations which will be used in the sequel. Then we will develop a more general mathematical frame with boundary distributions in Sobolev spaces in order to obtain diffraction solutions satisfying physical energy conditions.

Monochromatic light waves with a fixed wavenumber ๐‘˜>0 satisfy the first-order system of Maxwell-Helmholtz equations๐‘–๐‘˜๐ธโˆ—๎€ท๐‘ฅ๎€ธ+โˆ‡ร—๐ตโˆ—๎€ท๐‘ฅ๎€ธ=0,โˆ’๐‘–๐‘˜๐ตโˆ—๎€ท๐‘ฅ๎€ธ+โˆ‡ร—๐ธโˆ—๎€ท๐‘ฅ๎€ธ=0.(2.1) In the whole paper we consider a real wavenumber ๐‘˜>0, although the results can be generalised to the case of a complex wavenumber ๐‘˜โ‰ 0 with Re๐‘˜โ‰ฅ0 and Im๐‘˜โ‰ฅ0. We assume that the electromagnetic field with components ๐‘’๐‘—,๐‘๐‘—, ๐‘—=1,2,3,โŽ›โŽœโŽœโŽœโŽœโŽ๐‘’1๎€ท๐‘ฅ,๐‘ฅ3๎€ธ๐‘’2๎€ท๐‘ฅ,๐‘ฅ3๎€ธ๐‘’3๎€ท๐‘ฅ,๐‘ฅ3๎€ธโŽžโŽŸโŽŸโŽŸโŽŸโŽ =๐ธโˆ—๎€ท๐‘ฅ,๐‘ฅ3๎€ธ,โŽ›โŽœโŽœโŽœโŽœโŽ๐‘1๎€ท๐‘ฅ,๐‘ฅ3๎€ธ๐‘2๎€ท๐‘ฅ,๐‘ฅ3๎€ธ๐‘3๎€ท๐‘ฅ,๐‘ฅ3๎€ธโŽžโŽŸโŽŸโŽŸโŽŸโŽ =๐ตโˆ—๎€ท๐‘ฅ,๐‘ฅ3๎€ธ,(2.2) consists of functions defined in the upper half-space๎€ฝ๐‘ฅโ„‹โˆถ==๎€ท๐‘ฅ,๐‘ฅ3๎€ธโˆˆโ„3โˆฃ๐‘ฅโˆˆโ„2,๐‘ฅ3๎€พ.>0(2.3)

The diffraction problem is considered for an open bounded aperture๎€ฝฮฉโŠ‚(๐‘ฅ,0)โˆฃ๐‘ฅโˆˆโ„2๎€พ(2.4) in the screen plane ๐‘ฅ3=0. In the sequel we will suppress the notation of the third component 0 for the points in the screen plane, and interpret ฮฉ as well as the screen ฮฉ๐‘โˆถ=โ„2โงตฮฉ as subsets of โ„2. For describing the whole electromagnetic field in terms of its boundary values, for the moment we assume that these are functions, given for ๐‘ฅโˆˆโ„2 byโŽ›โŽœโŽœโŽœโŽœโŽ๐‘’1,0๐‘’(๐‘ฅ)2,0(๐‘’๐‘ฅ)3,0โŽžโŽŸโŽŸโŽŸโŽŸโŽ (๐‘ฅ)=lim๐‘ฅ3โ†“0๐ธโˆ—๎€ท๐‘ฅ,๐‘ฅ3๎€ธ,โŽ›โŽœโŽœโŽœโŽœโŽ๐‘1,0๐‘(๐‘ฅ)2,0(๐‘๐‘ฅ)3,0โŽžโŽŸโŽŸโŽŸโŽŸโŽ (๐‘ฅ)=lim๐‘ฅ3โ†“0๐ตโˆ—๎€ท๐‘ฅ,๐‘ฅ3๎€ธ.(2.5)

The screen ฮฉ๐‘ is assumed to be an ideal conducting wall. This implies the physical boundary conditions๐‘’1,0(๐‘ฅ)=๐‘’2,0(๐‘ฅ)=0โˆ€๐‘ฅโˆˆฮฉ๐‘.(2.6) In the general case the boundary fields (2.5) have to be replaced by appropriate distributions, and the limit ๐‘ฅ3โ†“0 will be performed in the Fourier domain instead of the half-space โ„‹.

For this purpose we write again ๐‘ฅ=(๐‘ฅ,๐‘ฅ3), with ๐‘ฅ=(๐‘ฅ1,๐‘ฅ2)โˆˆโ„2 and fixed ๐‘ฅ3>0, and assume that each field component ๐‘ข(โ‹…,๐‘ฅ3) represents a tempered distribution in ๐’ฎโ€ฒ(โ„2) with Fourier transform๎€ท๐œ‰ฬ‚๐‘ข1,๐œ‰2,๐‘ฅ3๎€ธ=1๎€2๐œ‹โ„2๐‘ข๎€ท๐‘ฅ1,๐‘ฅ2,๐‘ฅ3๎€ธ๐‘’โˆ’๐‘–(๐œ‰1๐‘ฅ1+๐œ‰2๐‘ฅ2)๐‘‘๐‘ฅ1๐‘‘๐‘ฅ2,๐œ‰1,๐œ‰2โˆˆโ„.(2.7) Then we obtain from the first-order Maxwell-Helmholtz equations the following Fourier transformed Maxwell-Helmholtz system: for all ๐‘ฅ3โ‰ฅ0 and fixed ๐œ‰1,๐œ‰2โˆˆโ„ we haveโŽ›โŽœโŽœโŽœโŽœโŽ๐‘–๐‘˜ฬ‚๐‘’1ฬ‚๐‘’2ฬ‚๐‘’3โŽžโŽŸโŽŸโŽŸโŽŸโŽ +โŽ›โŽœโŽœโŽœโŽœโŽœโŽ๐‘–๐œ‰2ฬ‚๐‘3โˆ’๐‘‘๐‘‘๐‘ฅ3ฬ‚๐‘2๐‘‘๐‘‘๐‘ฅ3ฬ‚๐‘1โˆ’๐‘–๐œ‰1ฬ‚๐‘3๐‘–๐œ‰1ฬ‚๐‘2โˆ’๐‘–๐œ‰2ฬ‚๐‘1โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽ000โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,โŽ›โŽœโŽœโŽœโŽœโŽฬ‚๐‘โˆ’๐‘–๐‘˜1ฬ‚๐‘2ฬ‚๐‘3โŽžโŽŸโŽŸโŽŸโŽŸโŽ +โŽ›โŽœโŽœโŽœโŽœโŽœโŽ๐‘–๐œ‰2ฬ‚๐‘’3โˆ’๐‘‘๐‘‘๐‘ฅ3ฬ‚๐‘’2๐‘‘๐‘‘๐‘ฅ3ฬ‚๐‘’1โˆ’๐‘–๐œ‰1ฬ‚๐‘’3๐‘–๐œ‰1ฬ‚๐‘’2โˆ’๐‘–๐œ‰2ฬ‚๐‘’1โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽ000โŽžโŽŸโŽŸโŽŸโŽŸโŽ .(2.8) Here we have replaced the partial derivative with respect to ๐‘ฅ3 by the ordinary derivative ๐‘‘/๐‘‘๐‘ฅ3.

We define1๐ถโˆถ=๐‘˜โŽ›โŽœโŽœโŽœโŽœโŽœโŽ๐œ•2๐œ•๐‘ฅ1๐œ•๐‘ฅ2โˆ’๎ƒฉ๐‘˜2+๐œ•2๐œ•๐‘ฅ21๎ƒช๐‘˜2+๐œ•2๐œ•๐‘ฅ22โˆ’๐œ•2๐œ•๐‘ฅ1๐œ•๐‘ฅ2โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(2.9) With the two-dimensional Laplace operator ฮ” we have๐ถ2๎€ท๐‘˜=โˆ’2๎€ธโŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ +ฮ”1001.(2.10) For the action of ๐ถ in the Fourier domain we obtain multiplication by the matrix๎1๐ถ(๐œ‰)โˆถ=๐‘˜โŽ›โŽœโŽœโŽโˆ’๐œ‰1๐œ‰2โˆ’๎€ท๐‘˜2โˆ’๐œ‰21๎€ธ๐‘˜2โˆ’๐œ‰22๐œ‰1๐œ‰2โŽžโŽŸโŽŸโŽ ๎€ท๐œ‰,๐œ‰=1,๐œ‰2๎€ธโˆˆโ„2.(2.11) We replace (2.8) with the ordinary differential equations๐‘‘๐‘‘๐‘ฅ3โŽ›โŽœโŽœโŽฬ‚๐‘’1ฬ‚๐‘’2โŽžโŽŸโŽŸโŽ ๎๐ถโŽ›โŽœโŽœโŽฬ‚๐‘=โˆ’๐‘–1ฬ‚๐‘2โŽžโŽŸโŽŸโŽ ,๐‘‘๐‘‘๐‘ฅ3โŽ›โŽœโŽœโŽฬ‚๐‘1ฬ‚๐‘2โŽžโŽŸโŽŸโŽ ๎๐ถโŽ›โŽœโŽœโŽ=๐‘–ฬ‚๐‘’1ฬ‚๐‘’2โŽžโŽŸโŽŸโŽ ,(2.12) and the two algebraic conditionsฬ‚๐‘’3=1๐‘˜๎€ท๐œ‰2ฬ‚๐‘1โˆ’๐œ‰1ฬ‚๐‘2๎€ธ,ฬ‚๐‘3=1๐‘˜๎€ท๐œ‰1ฬ‚๐‘’2โˆ’๐œ‰2ฬ‚๐‘’1๎€ธ.(2.13) For all fixed ๐œ‰โˆˆโ„2 we supplement the system of differential equations (2.12) by the initial conditionsฬ‚๐‘’๐‘—,0(๐œ‰)โˆถ=ฬ‚๐‘’๐‘—ฬ‚๐‘(๐œ‰,0),๐‘—,0ฬ‚๐‘(๐œ‰)โˆถ=๐‘—(๐œ‰,0),๐‘—=1,2,(2.14) and put โŽงโŽชโŽจโŽชโŽฉ๎”๐‘š(๐œ‰)โˆถ=๐‘˜2โˆ’||๐œ‰||2,||๐œ‰||๐‘–๎”โ‰ค๐‘˜,||๐œ‰||2โˆ’๐‘˜2,||๐œ‰||>๐‘˜.(2.15) Then the general solution of the homogeneous linear system (2.12) isโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽฬ‚๐‘’1๎€ท๐œ‰,๐‘ฅ3๎€ธฬ‚๐‘’2๎€ท๐œ‰,๐‘ฅ3๎€ธฬ‚๐‘1๎€ท๐œ‰,๐‘ฅ3๎€ธฬ‚๐‘2๎€ท๐œ‰,๐‘ฅ3๎€ธโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽ๎€ทcos๐‘š(๐œ‰)๐‘ฅ3๎€ธ๎€ท๐ธโˆ’๐‘–sin๐‘š(๐œ‰)๐‘ฅ3๎€ธ๎๐‘–๎€ท๐‘š(๐œ‰)๐ถ(๐œ‰)sin๐‘š(๐œ‰)๐‘ฅ3๎€ธ๎๐ถ๎€ท๐‘š๐‘š(๐œ‰)(๐œ‰)cos(๐œ‰)๐‘ฅ3๎€ธ๐ธโŽžโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽฬ‚๐‘’1,0(๐œ‰)ฬ‚๐‘’2,0(ฬ‚๐‘๐œ‰)1,0ฬ‚๐‘(๐œ‰)2,0โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ (๐œ‰)(2.16) with the 2ร—2 unit matrix ๐ธ.

But the terms cos(๐‘š(๐œ‰)๐‘ฅ3) and sin(๐‘š(๐œ‰)๐‘ฅ3) are exponentially increasing for fixed ๐‘ฅ3>0 and |๐œ‰|โ†’โˆž. For avoiding that the Fourier transformed fields are also exponentially increasing we have to require the following algebraic conditions for |๐œ‰|>๐‘˜:โŽ›โŽœโŽœโŽฬ‚๐‘๐‘šโ‹…1,0ฬ‚๐‘2,0โŽžโŽŸโŽŸโŽ =๎๐ถโŽ›โŽœโŽœโŽฬ‚๐‘’1,0ฬ‚๐‘’2,0โŽžโŽŸโŽŸโŽ .(2.17) By using (2.17), for ๐‘ฅ3โ†“0 and |๐œ‰|>๐‘˜ we can replace (2.13) withฬ‚๐‘’3,01=โˆ’๐‘š๎€ท๐œ‰1ฬ‚๐‘’1,0+๐œ‰2ฬ‚๐‘’2,0๎€ธ,ฬ‚๐‘3,01=โˆ’๐‘˜๎€ท๐œ‰2ฬ‚๐‘’1,0โˆ’๐œ‰1ฬ‚๐‘’2,0๎€ธ.(2.18) From the general solution and (2.17), (2.18) we obtain the following decay conditions for ๐‘ฅ3>0:ฬ‚๐‘’๐‘—๎€ท๐œ‰,๐‘ฅ3๎€ธ=๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)ฬ‚๐‘’๐‘—,0ฬ‚๐‘(๐œ‰),๐‘—๎€ท๐œ‰,๐‘ฅ3๎€ธ=๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)ฬ‚๐‘๐‘—,0(๐œ‰),๐‘—=1,2,3.(2.19) With ๐‘ฅ=(๐‘ฅ1,๐‘ฅ2)โˆˆโ„2, ๐‘ฅ=(๐‘ฅ,๐‘ฅ3)โˆˆโ„3 and ๐‘ฅ3>0 there holds the important and well-known Sommerfeld-Weyl integral representation๐น๐‘˜๎€ท๐‘ฅ,๐‘ฅ3๎€ธ๐‘’โˆถ=๐‘–๐‘˜|๐‘ฅ|||๐‘ฅ||=๐‘–๎€2๐œ‹โ„2๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘š(๐œ‰)๐‘‘๐œ‰.(2.20) The left-hand side in (2.20) is the singular solution of the three-dimensional Helmholtz equation (ฮ”+๐‘˜2)๐น๐‘˜=โˆ’4๐œ‹๐›ฟ. For this reason it is natural to require the algebraic conditions (2.17), (2.18) also in the case |๐œ‰|<๐‘˜, such that (2.19) is generally valid for ๐‘ฅ3>0 and ๐œ‰โˆˆโ„2, |๐œ‰|โ‰ ๐‘˜.

Distributions ๐‘ขโˆˆ๐’Ÿโ€ฒ(โ„2) with compact support in the screen plane are tempered, and it follows from the Paley-Wiener theorem that ฬ‚๐‘ข is a smooth function which has polynomial growth on โ„2. This is used in the following theorem, which results if we regard (2.17), (2.18), and (2.19) and apply the Fourier inversion formula for ๐‘ฅ3>0 to each component ฬ‚๐‘’๐‘—(โ‹…,๐‘ฅ3) and ฬ‚๐‘๐‘—(โ‹…,๐‘ฅ3).

Theorem 2.1. Let there be given ๐‘’1,0,๐‘’2,0โˆˆ๐’ฎโ€ฒ(โ„2) with support in the bounded region ฮฉ. Then the following functions ๐‘’๐‘—,๐‘๐‘—โˆถโ„‹โ†’โ„‚ constitute a ๐ถโˆž-solution of the Maxwell-Helmholtz system (2.1) in the upper half-space โ„‹, ๐‘—=1,2,3; ๐‘’1๎€ท๐‘ฅ,๐‘ฅ3๎€ธ=1๎€2๐œ‹โ„2ฬ‚๐‘’1,0(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘’๐‘‘๐œ‰,2๎€ท๐‘ฅ,๐‘ฅ3๎€ธ=1๎€2๐œ‹โ„2ฬ‚๐‘’2,0(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘๐‘‘๐œ‰,1๎€ท๐‘ฅ,๐‘ฅ3๎€ธ1=โˆ’๎€2๐œ‹๐‘˜โ„2๐œ‰1๐œ‰2ฬ‚๐‘’1,0๎€ท๐‘˜(๐œ‰)+2โˆ’๐œ‰21๎€ธฬ‚๐‘’2,0(๐œ‰)๐‘’๐‘š(๐œ‰)๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘๐‘‘๐œ‰,2๎€ท๐‘ฅ,๐‘ฅ3๎€ธ=1๎€2๐œ‹๐‘˜โ„2๎€ท๐‘˜2โˆ’๐œ‰22๎€ธฬ‚๐‘’1,0(๐œ‰)+๐œ‰1๐œ‰2ฬ‚๐‘’2,0(๐œ‰)๐‘’๐‘š(๐œ‰)๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘’๐‘‘๐œ‰,3๎€ท๐‘ฅ,๐‘ฅ3๎€ธ1=โˆ’๎€2๐œ‹โ„2๐œ‰1ฬ‚๐‘’1,0(๐œ‰)+๐œ‰2ฬ‚๐‘’2,0(๐œ‰)๐‘š๐‘’(๐œ‰)๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘๐‘‘๐œ‰,3๎€ท๐‘ฅ,๐‘ฅ3๎€ธ1=โˆ’๎€2๐œ‹๐‘˜โ„2๎€บ๐œ‰2ฬ‚๐‘’1,0(๐œ‰)โˆ’๐œ‰1ฬ‚๐‘’2,0๎€ป๐‘’(๐œ‰)๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰.(2.21)

Proof. The calculation of the partial derivatives of ๐‘’๐‘— and ๐‘๐‘— can be interchanged with integration. This can be used to check the Maxwell-Helmholtz equations independent from the previous representations of the Fourier-transforms ฬ‚๐‘’๐‘— and ฬ‚๐‘๐‘— in terms of ฬ‚๐‘’1,0 and ฬ‚๐‘’2,0.

Definition 2.2. The electromagnetic field in the half-space ๐‘ฅ3>0 behind the screen is completely determined by the electric boundary components ๐‘’1,0, ๐‘’2,0. We call ๐‘’๐‘—,๐‘๐‘—โˆถโ„‹โ†’โ„‚, ๐‘—=1,2,3, the half-space solution determined by the boundary distributions ๐‘’1,0,๐‘’2,0โˆˆ๐’ฎโ€ฒ(โ„2) with compact support in ฮฉ.

Remark 2.3. Assume that ๐‘’1,0,๐‘’2,0 are smooth functions with compact support in ฮฉ and that ๐ธโˆ—, ๐ตโˆ— is the corresponding electromagnetic field with the components given in Theorem 2.1. Then we obtain from Fourierโ€™s inversion formula that for all ๐‘ฅโˆˆโ„2lim๐‘ฅ3โ†“0๐‘’๐‘—๎€ท๐‘ฅ,๐‘ฅ3๎€ธ=๐‘’๐‘—,0(๐‘ฅ),๐‘—=1,2.(2.22)

Remark 2.4. Assume again that ๐‘’1,0,๐‘’2,0 are smooth functions with compact support in ฮฉ. Then we obtain from Theorem 2.1 and the Sommerfeld-Weyl integral (2.20) for all ๐‘ฅโˆˆโ„2 and all ๐‘ฅ3>0 that ๐‘1๎€ท๐‘ฅ,๐‘ฅ3๎€ธ๐‘–=โˆ’๐œ•2๐œ‹๐‘˜2๐œ•๐‘ฅ1๐œ•๐‘ฅ2๎€ฮฉ๐‘’1,0(๐‘ฆ)๐น๐‘˜๎€ท๐‘ฅโˆ’๐‘ฆ,๐‘ฅ3๎€ธ+๐‘–๐‘‘๐‘ฆ๎ƒฉ๐‘˜2๐œ‹๐‘˜2+๐œ•2๐œ•๐‘ฅ21๎ƒช๎€ฮฉ๐‘’2,0(๐‘ฆ)๐น๐‘˜๎€ท๐‘ฅโˆ’๐‘ฆ,๐‘ฅ3๎€ธ๐‘๐‘‘๐‘ฆ,2๎€ท๐‘ฅ,๐‘ฅ3๎€ธ๐‘–=โˆ’๎ƒฉ๐‘˜2๐œ‹๐‘˜2+๐œ•2๐œ•๐‘ฅ22๎ƒช๎€ฮฉ๐‘’1,0(๐‘ฆ)๐น๐‘˜๎€ท๐‘ฅโˆ’๐‘ฆ,๐‘ฅ3๎€ธ+๐‘–๐‘‘๐‘ฆ๐œ•2๐œ‹๐‘˜2๐œ•๐‘ฅ1๐œ•๐‘ฅ2๎€ฮฉ๐‘’2,0(๐‘ฆ)๐น๐‘˜๎€ท๐‘ฅโˆ’๐‘ฆ,๐‘ฅ3๎€ธ๐‘‘๐‘ฆ.(2.23) Equations (2.23) involve the boundary conditions (2.6) for the electric field components on the ideal conducting plane screen ฮฉ๐‘. In order to obtain a coupled system of boundary integro-differential equations we pass for ๐‘ฅโˆˆโ„2 to the limit ๐‘ฅ3โ†“0 and obtain ๐‘1,0๐‘–(๐‘ฅ)=โˆ’๐œ•2๐œ‹๐‘˜2๐œ•๐‘ฅ1๐œ•๐‘ฅ2๎€ฮฉ๐‘’1,0๐‘’(๐‘ฆ)๐‘–๐‘˜|๐‘ฅโˆ’๐‘ฆ|||||+๐‘–๐‘ฅโˆ’๐‘ฆ๐‘‘๐‘ฆ๎ƒฉ๐‘˜2๐œ‹๐‘˜2+๐œ•2๐œ•๐‘ฅ21๎ƒช๎€ฮฉ๐‘’2,0๐‘’(๐‘ฆ)๐‘–๐‘˜|๐‘ฅโˆ’๐‘ฆ|||||๐‘๐‘ฅโˆ’๐‘ฆ๐‘‘๐‘ฆ,2,0๐‘–(๐‘ฅ)=โˆ’๎ƒฉ๐‘˜2๐œ‹๐‘˜2+๐œ•2๐œ•๐‘ฅ22๎ƒช๎€ฮฉ๐‘’1,0๐‘’(๐‘ฆ)๐‘–๐‘˜|๐‘ฅโˆ’๐‘ฆ|||||+๐‘–๐‘ฅโˆ’๐‘ฆ๐‘‘๐‘ฆ๐œ•2๐œ‹๐‘˜2๐œ•๐‘ฅ1๐œ•๐‘ฅ2๎€ฮฉ๐‘’2,0(๐‘’๐‘ฆ)๐‘–๐‘˜|๐‘ฅโˆ’๐‘ฆ|||||๐‘ฅโˆ’๐‘ฆ๐‘‘๐‘ฆ.(2.24) In general the electric boundary fields ๐‘’1,0 and ๐‘’2,0 are unknown distributions with compact support in ฮฉ, whereas ๐‘1,0 and ๐‘2,0 are given distributions in the aperture ฮฉ. In order to select physical admissible solutions of the diffraction problem we need some conditions for its electromagnetic energy content, especially in local volume elements ๐บโŠ‚โ„‹. Recall that โ„‹โˆถ=โ„2ร—โ„+.

Definition 2.5. Let ๐‘’๐‘—,๐‘๐‘—โˆถโ„‹โ†’โ„‚, ๐‘—=1,2,3, be the half-space solution determined by the boundary distributions ๐‘’1,0,๐‘’2,0โˆˆ๐’ฎโ€ฒ(โ„2) with compact support in ฮฉ.(a)The solution is called physical admissible if and only if it satisfies the local energy condition123๎“๐‘—=1๎€ž๐บ๎‚€||๐‘’๐‘—๎€ท๐‘ฅ๎€ธ||2+||๐‘๐‘—๎€ท๐‘ฅ๎€ธ||2๎‚๐‘‘๐‘ฅ<โˆž(2.25)for every bounded domain ๐บโŠ‚โ„‹.(b)The solution satisfies the stronger global energy condition if and only if 123๎“๐‘—=1๎€žโ„’โ„Ž๎‚€||๐‘’๐‘—๎€ท๐‘ฅ๎€ธ||2+||๐‘๐‘—๎€ท๐‘ฅ๎€ธ||2๎‚๐‘‘๐‘ฅ<โˆž(2.26)for some โ„Ž>0, and therewith for all โ„Ž>0, with the layerโ„’โ„Žโˆถ={(๐‘ฅ1,๐‘ฅ2,๐‘ฅ3)โˆˆโ„‹|๐‘ฅ3<โ„Ž}.
In the following two sections we determine the solutions with global as well as those with local finite energy in terms of an appropriate functional analytical setting for the electric boundary fields ๐‘’1,0 and ๐‘’2,0.

3. The Global Energy Condition

Throughout the rest of this paper let the open aperture โ‹ƒฮฉโˆถ=๐œ…๐‘—=1ฮฉ๐‘—โŠ‚โ„2 be a finite union of nonempty bounded Lipschitz domains ฮฉ๐‘— in the screen plane, such that the compact sets ฮฉ๐‘— are pairwise disjoint.

By ๐ป๐‘ (โ„2), ๐‘ โˆˆโ„, we denote the Sobolev space of tempered distributions โ„Ž, for which the Fourier transform ๎โ„Ž is locally integrable withโ€–โ„Žโ€–๐ป๐‘ (โ„2)๎‚ต๎€โˆถ=โ„2|||๎|||โ„Ž(๐œ‰)2๎‚€||๐œ‰||1+2๎‚๐‘ ๎‚ถ๐‘‘๐œ‰1/2<โˆž(3.1) (cf. [3], Chapterโ€‰โ€‰8.8). โ€–โ‹…โ€–๐ป๐‘ (โ„2) is the norm on the Banach space ๐ป๐‘ (โ„2).

For ๐‘ โˆˆโ„ the Sobolev space ๎‚๐ป๐‘ (ฮฉ) is given by๎‚๐ป๐‘ ๎‚†(ฮฉ)=โ„Žโˆˆ๐ป๐‘ ๎€ทโ„2๎€ธโˆฃsuppโ„ŽโŠ‚ฮฉ๎‚‡.(3.2) Here suppโ„Ž denotes the support of the distribution โ„Ž in the compact set ฮฉ. The space ๎‚๐ป๐‘ (ฮฉ) is equipped with the norm of ๐ป๐‘ (โ„2), which makes it into a Banach space (cf. [4, Chapterโ€‰โ€‰3]). The Lipschitz property of ๐œ•ฮฉ guarantees that ๎‚๐ป๐‘ (ฮฉ) is the closure of ๐’Ÿ(ฮฉ) in ๐ป๐‘ (โ„2). For a more general result see [4, Theoremโ€‰โ€‰3.29].

Theorem 3.1. Let ๐‘’๐‘—,๐‘๐‘—โˆถโ„‹โ†’โ„‚, ๐‘—=1,2,3, be the half-space solution determined by the boundary distributions ๐‘’1,0,๐‘’2,0โˆˆ๐’ฎโ€ฒ(โ„2) with compact support in ฮฉ. Then the diffraction solution has global finite energy if and only if๐‘’1,0,๐‘’2,0โŽ›โŽœโŽœโŽ,โˆ‡โ‹…โˆ’๐‘’2,0๐‘’1,0โŽžโŽŸโŽŸโŽ โˆˆ๎‚๐ปโˆ’1/2(ฮฉ)(3.3) and ฬ‚๐‘’1,0(๐œ‰)๐œ‰1+ฬ‚๐‘’2,0(๐œ‰)๐œ‰2=0 for all ๐œ‰=(๐œ‰1,๐œ‰2)โˆˆโ„2 with |๐œ‰|=๐‘˜.

Proof. In order to perform the energy evaluation with Parsevalโ€™s theorem we define with the Fourier transformed electromagnetic boundary fields ฬ‚๐‘’๐‘—,0,ฬ‚๐‘๐‘—,0, ๐‘—=1,2,3, the quantity ๐‘Š01(๐œ‰)โˆถ=23๎“๐‘—=1๎‚€||ฬ‚๐‘’๐‘—,0||(๐œ‰)2+||ฬ‚๐‘๐‘—,0||(๐œ‰)2๎‚.(3.4) It follows from a lengthy calculation with |๐‘ง|2=๐‘ง๐‘ง for the values ๐‘งโˆˆโ„‚ of the Fourier transformed boundary fields as well as from the algebraic relations (2.17), (2.18), which are valid for |๐œ‰|โ‰ ๐‘˜, that ๐‘Š0๎ƒฏ||(๐œ‰)=ฬ‚๐‘’1,0||(๐œ‰)2+||ฬ‚๐‘’2,0||(๐œ‰)2+||ฬ‚๐‘’3,0||(๐œ‰)2,||๐œ‰||||<๐‘˜,ฬ‚๐‘’3,0||(๐œ‰)2+||ฬ‚๐‘3,0||(๐œ‰)2,||๐œ‰||>๐‘˜.(3.5)
With ๐‘‰1โˆถ={๐œ‰โˆˆโ„2||๐œ‰|<๐‘˜}, ๐‘‰2โˆถ={๐œ‰โˆˆโ„2||๐œ‰|>๐‘˜} we conclude from Parsevalโ€™s theorem, (2.19) and the definition (2.15) of ๐‘š(๐œ‰) for โ„Ž>01โ„ฐ(โ„Ž)โˆถ=23๎“๐‘—=1๎€œโ„Ž0๎€โ„2๎‚€||๐‘’๐‘—๎€ท๐‘ฅ,๐‘ฅ3๎€ธ||2+||๐‘๐‘—๎€ท๐‘ฅ,๐‘ฅ3๎€ธ||2๎‚๐‘‘๐‘ฅ๐‘‘๐‘ฅ3๎€=โ„Ž๐‘‰1๐‘Š0(๎€๐œ‰)๐‘‘๐œ‰+๐‘‰21โˆ’๐‘’โˆšโˆ’2โ„Ž|๐œ‰|2โˆ’๐‘˜22๎”||๐œ‰||2โˆ’๐‘˜2๐‘Š0(๐œ‰)๐‘‘๐œ‰.(3.6) Using for all ๐œ‰โˆˆโ„2 with |๐œ‰|โ‰ ๐‘˜ the estimate 12๎‚€||ฬ‚๐‘’1,0||(๐œ‰)2+||ฬ‚๐‘’2,0||(๐œ‰)2+||ฬ‚๐‘’3,0||(๐œ‰)2+||ฬ‚๐‘3,0||(๐œ‰)2๎‚โ‰ค๐‘Š0||(๐œ‰)โ‰คฬ‚๐‘’1,0||(๐œ‰)2+||ฬ‚๐‘’2,0||(๐œ‰)2+||ฬ‚๐‘’3,0||(๐œ‰)2+||ฬ‚๐‘3,0||(๐œ‰)2,(3.7) and observing that we have uniformly in ๐‘Ÿ>๐‘˜ for appropriate constants ๐›ผ,๐›ฝ>0๐›ผโˆš1+๐‘Ÿ2โ‰ค1โˆ’๐‘’โˆšโˆ’2โ„Ž๐‘Ÿ2โˆ’๐‘˜22โˆš๐‘Ÿ2โˆ’๐‘˜2โ‰ค๐›ฝโˆš1+๐‘Ÿ2,(3.8) we conclude from (3.6) that โ„ฐ(โ„Ž) is finite if and only if ๐‘’1,0,๐‘’2,0,๐‘’3,0,๐‘3,0=๐‘–๐‘˜โŽ›โŽœโŽœโŽโˆ‡โ‹…โˆ’๐‘’2,0๐‘’1,0โŽžโŽŸโŽŸโŽ โˆˆ๐ปโˆ’1/2๎€ทโ„2๎€ธ.(3.9) Note that with ๐‘’1,0 and ๐‘’2,0 also โˆ‡โ‹…(โˆ’๐‘’2,0,๐‘’1,0)๐‘‡ has support in ฮฉ.
Assume first that โ„ฐ(โ„Ž)<โˆž and that ฬ‚๐‘’1,0(๐œ‰)๐œ‰1+ฬ‚๐‘’2,0(๐œ‰)๐œ‰2โ‰ 0 for a certain ๐œ‰=(๐œ‰1,๐œ‰2)โˆˆโ„2 with |๐œ‰|=๐‘˜. Since ๐‘’1,0,๐‘’2,0 have compact support, we conclude from the Paley-Wiener theorem that ฬ‚๐‘’1,0,ฬ‚๐‘’2,0โˆถโ„2โ†’โ„‚ are smooth and especially continuous. Thus |ฬ‚๐‘’1,0(๐œ‰)๐œ‰1+ฬ‚๐‘’2,0(๐œ‰)๐œ‰2|2โ‰ฅ๐›ฟ>0 in a bounded domain ๎€ท๐œ‰1,๐œ‰2๎€ธ=๐‘Ÿ(cos๐œ‘,sin๐œ‘)with๐œ‘1<๐œ‘<๐œ‘2,๐‘˜<๐‘Ÿ<๐‘˜+๐œ€,(3.10) and hence |ฬ‚๐‘’3,0|2 because of (2.18) is not integrable. Due to (3.1) this violates the necessary condition ๐‘’3,0โˆˆ๐ปโˆ’1/2(โ„2) in (3.9).
For showing the other direction of the equivalence stated in the theorem we assume that ๐‘’1,0,๐‘’2,0โŽ›โŽœโŽœโŽ,โˆ‡โ‹…โˆ’๐‘’2,0๐‘’1,0โŽžโŽŸโŽŸโŽ โˆˆ๎‚๐ปโˆ’1/2(ฮฉ),(3.11)ฬ‚๐‘’1,0๎€ท๐œ‰1,๐œ‰2๎€ธ๐œ‰1+ฬ‚๐‘’2,0๎€ท๐œ‰1,๐œ‰2๎€ธ๐œ‰2=0,(3.12) for all ๐œ‰1,๐œ‰2โˆˆโ„ with ๐œ‰21+๐œ‰22=๐‘˜2. In order to prove (3.9) it remains to show that ๐‘’3,0โˆˆ๐ปโˆ’1/2(โ„2). Since ๐‘’1,0, ๐‘’2,0 have compact support in ฮฉ, we obtain from the Paley-Wiener theorem that the Fourier transforms ฬ‚๐‘’1,0, ฬ‚๐‘’2,0 can be continuated to entire functions. We also denote these entire functions by ฬ‚๐‘’1,0,ฬ‚๐‘’2,0. We define the entire function ๐‘“ by ๐‘“๎€ท๐‘ง1,๐‘ง2๎€ธโˆถ=ฬ‚๐‘’1,0๎€ท๐‘ง1,๐‘ง2๎€ธ๐‘ง1+ฬ‚๐‘’2,0๎€ท๐‘ง1,๐‘ง2๎€ธ๐‘ง2.(3.13) Using (3.12) and (2.18), it follows for example from Theorem A.1 in the appendix that |ฬ‚๐‘’3,0|2 is a locally integrable function.
From the first equation in (2.18) and the Cauchy-Schwarz inequality we obtain the estimate ||ฬ‚๐‘’3,0||(๐œ‰)2โ‰ค||๐œ‰||2||๐œ‰||2โˆ’๐‘˜2โ‹…๎‚€||ฬ‚๐‘’1,0||(๐œ‰)2+||ฬ‚๐‘’2,0||(๐œ‰)2๎‚,||๐œ‰||>๐‘˜.(3.14) Now ๐‘’3,0โˆˆ๐ปโˆ’1/2(โ„2) follows from the assumption ๐‘’1,0,๐‘’2,0โˆˆ๎‚๐ปโˆ’1/2(ฮฉ), because |๐‘’3,0|2 is locally integrable and |๐œ‰|2/(|๐œ‰|2โˆ’๐‘˜2) is bounded for sufficiently large |๐œ‰|.

Next we present a characterisation of the global finite energy solutions in terms of two boundary potential functions. In the following we use the vectorial differential operator (2.9) satisfying (2.10).

Theorem 3.2. Boundary potential functions(a)Given are two functions ๐‘ข1,๐‘ข2โˆˆ๎‚๐ป1/2(ฮฉ), in the following called boundary potential functions, satisfying the regularity condition โŽ›โŽœโŽœโŽโˆ‡โ‹…โˆ’๐‘ข2๐‘ข1โŽžโŽŸโŽŸโŽ โˆˆ๎‚๐ป1/2(ฮฉ).(3.15)Define โŽ›โŽœโŽœโŽ๐‘’1,0๐‘’2,0โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐‘ขโˆถ=๐ถ1๐‘ข2โŽžโŽŸโŽŸโŽ .(3.16)Then the corresponding electromagnetic field ๐‘’๐‘—,๐‘๐‘—, ๐‘—=1,2,3, determined by ๐‘’1,0 and ๐‘’2,0 according to Theorem 2.1, has global finite energy. (b)Let ๐‘’1,0, ๐‘’2,0 be given such that the half-space solution in Theorem 2.1 has global finite energy. If the set โ„2โงตฮฉ is connected, then the boundary fields ๐‘’1,0 and ๐‘’2,0 can be represented by two boundary potential functions ๐‘ข1,๐‘ข2โˆˆ๎‚๐ป1/2(ฮฉ) satisfying (3.15) and (3.16) in part (a).(c)From the assumptions of part (a), or alternatively part (b), one obtains in the distributional sense ๐‘1,0๐‘–(๐‘ฅ)=๎€ท๐‘˜2๐œ‹2๎€ธ๎€+ฮ”ฮฉ๐‘ข1๐‘’(๐‘ฆ)๐‘–๐‘˜|๐‘ฅโˆ’๐‘ฆ|||||๐‘๐‘ฅโˆ’๐‘ฆ๐‘‘๐‘ฆ,2,0๐‘–(๐‘ฅ)=๎€ท๐‘˜2๐œ‹2๎€ธ๎€+ฮ”ฮฉ๐‘ข2๐‘’(๐‘ฆ)๐‘–๐‘˜|๐‘ฅโˆ’๐‘ฆ|||||๐‘ฅโˆ’๐‘ฆ๐‘‘๐‘ฆ(3.17)on the whole screen plane โ„2 with ๐‘1,0,๐‘2,0โŽ›โŽœโŽœโŽ,โˆ‡โ‹…โˆ’๐‘2,0๐‘1,0โŽžโŽŸโŽŸโŽ โˆˆ๐ปโˆ’1/2๎€ทโ„2๎€ธ.(3.18)

Proof. Part (a) follows in the Fourier domain by representing ฬ‚๐‘’1,0, ฬ‚๐‘’2,0, ฬ‚๐‘’3,0 and ฬ‚๐‘3,0 in terms of ฬ‚๐‘ข1 and ฬ‚๐‘ข2 as ฬ‚๐‘’1,0=โˆ’๐‘˜ฬ‚๐‘ข2โˆ’๐œ‰1๐‘˜๎€ท๐œ‰2ฬ‚๐‘ข1โˆ’๐œ‰1ฬ‚๐‘ข2๎€ธ,ฬ‚๐‘’2,0=+๐‘˜ฬ‚๐‘ข1โˆ’๐œ‰2๐‘˜๎€ท๐œ‰2ฬ‚๐‘ข1โˆ’๐œ‰1ฬ‚๐‘ข2๎€ธ,ฬ‚๐‘’3,0๐‘š=โˆ’๐‘˜๎€ท๐œ‰2ฬ‚๐‘ข1โˆ’๐œ‰1ฬ‚๐‘ข2๎€ธ,ฬ‚๐‘3,0=๐œ‰1ฬ‚๐‘ข1+๐œ‰2ฬ‚๐‘ข2.(3.19) These equations and the assumptions imply that ๐‘’1,0,๐‘’2,0,๐‘3,0โˆˆ๐ปโˆ’1/2(โ„2), and regarding โˆš|๐‘š(๐œ‰)|โ‰ค๐›ผ1+|๐œ‰|2 uniformly in ๐œ‰โˆˆโ„2 for a certain constant ๐›ผ>0 also ๐‘’3,0โˆˆ๐ปโˆ’1/2(โ„2). Since ๐‘’1,0, ๐‘’2,0 have compact support in ฮฉ like ๐‘ข1, ๐‘ข2, the proof of part (a) follows from the fact that condition (3.9) is equivalent to โ„ฐ(โ„Ž)<โˆž.
For proving part (b) we assume that the diffraction solution corresponding to ๐‘’1,0,๐‘’2,0โˆˆ๎‚๐ปโˆ’1/2(ฮฉ) has global finite energy. We conclude from Theorem 3.1 and Theorem A.1 in the appendix that we obtain entire functions ๐‘ฃ1,๐‘ฃ2โˆถโ„‚2โ†’โ„‚ by ๐‘ฃ1๎€ท๐œ‰1,๐œ‰2๎€ธ1=+๐‘˜ฬ‚๐‘’2,0๎€ท๐œ‰1,๐œ‰2๎€ธโˆ’๐œ‰2๐‘˜โ‹…ฬ‚๐‘’1,0(๐œ‰)๐œ‰1+ฬ‚๐‘’2,0(๐œ‰)๐œ‰2๐œ‰21+๐œ‰22โˆ’๐‘˜2,๐‘ฃ2๎€ท๐œ‰1,๐œ‰2๎€ธ1=โˆ’๐‘˜ฬ‚๐‘’1,0๎€ท๐œ‰1,๐œ‰2๎€ธ+๐œ‰1๐‘˜โ‹…ฬ‚๐‘’1,0(๐œ‰)๐œ‰1+ฬ‚๐‘’2,0(๐œ‰)๐œ‰2๐œ‰21+๐œ‰22โˆ’๐‘˜2.(3.20) From (3.20) we get for ๐‘—=1,2 that ๎€โ„2||๐‘ฃ๐‘—(||๐œ‰)2๎‚€||๐œ‰||1+2๎‚1/2๐‘‘๐œ‰<โˆž.(3.21) Thus for ๐‘—=1,2 the inverse Fourier transform ๐‘ข๐‘— of ๐‘ฃ๐‘— lies in ๐ป1/2(โ„2).
Now we show that ๐‘ข1 and ๐‘ข2 have their support in ฮฉ and hence, by reason of (3.21), ๐‘ข1,๐‘ข2โˆˆ๎‚๐ป1/2(ฮฉ). To this aim we choose some ๐‘…>0 such that ฮฉโŠ‚๐ต๐‘…๎€ฝโˆถ=๐‘ฅโˆˆโ„2๎€พ.โˆฃ|๐‘ฅ|<๐‘…(3.22) Since ๐‘’1,0,๐‘’2,0 are supported in ฮฉ, we have supp๐‘’1,0,supp๐‘’2,0โŠ‚๐ต๐‘…. Therefore we obtain from the Paley-Wiener theorem and (3.20) that ๐‘ข1 and ๐‘ข2 are also supported in ๐ต๐‘…, because ๐‘ฃ1, ๐‘ฃ2, as ฬ‚๐‘’1,0, ฬ‚๐‘’2,0, are of exponential type not larger than ๐‘….
Equations (3.20) can be resolved with respect to ฬ‚๐‘’1,0, ฬ‚๐‘’2,0. This gives (3.16). Together with (2.10) it follows that in the complement of ฮฉ there holds โˆ’๎€ท๐‘˜2๎€ธโŽ›โŽœโŽœโŽ๐‘ข+ฮ”1๐‘ข2โŽžโŽŸโŽŸโŽ =๐ถ2โŽ›โŽœโŽœโŽ๐‘ข1๐‘ข2โŽžโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐‘’=๐ถ1,0๐‘’2,0โŽžโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽ00โŽžโŽŸโŽŸโŽ .(3.23) Now we make use of the fact that the complement of ฮฉ is connected. Since supp๐‘ข๐‘—โŠ‚๐ต๐‘… for ๐‘—=1,2, Holmgrenโ€™s unique continuation principle, applied to the two scalar Helmholtz equations in (3.23), implies that ๐‘ข1, ๐‘ข2 have their support in ฮฉ.
Finally, since ๐‘ฃ๐‘—=ฬ‚๐‘ข๐‘— for ๐‘—=1,2, from (3.20) we obtain for๐œ‰โˆˆโ„2โˆ’๐œ‰1ฬ‚๐‘ข2+๐œ‰2ฬ‚๐‘ข1๐œ‰=โˆ’๐‘˜1ฬ‚๐‘’1,0+๐œ‰2ฬ‚๐‘’2,0๐œ‰21+๐œ‰22โˆ’๐‘˜2.(3.24) The asymptotic behaviour of this expression for |๐œ‰|โ†’โˆž shows the validity of the regularity condition (3.15).
The convolution integrals in part (c) can be rewritten in the Fourier domain by using the Sommerfeld-Weyl integral representation (2.20) in the limit ๐‘ฅ3โ†“0. The resulting relations in the Fourier domain follow from the representations of the components ๐‘1 and ๐‘2 in Theorem 2.1.
The validity of (3.18) is a consequence of the algebraic relations between ฬ‚๐‘ข1, ฬ‚๐‘ข2, ฬ‚๐‘1,0, and ฬ‚๐‘2,0.

Remark 3.3. Consider the Sobolev spaces ๐ป๐‘ (ฮฉ) given for ๐‘ โˆˆโ„ by ๐ป๐‘ ๎€ฝ(ฮฉ)=๐น|ฮฉโˆฃ๐นโˆˆ๐ป๐‘ ๎€ทโ„2,๎€ธ๎€พ(3.25) with the restriction ๐น|ฮฉ of the tempered distribution ๐นโˆถ๐’ฎ(โ„2)โ†’โ„‚ to the subspace ๐’ฎ(ฮฉ) of the Schwartz space ๐’ฎ(โ„2), where ๐’ฎ(ฮฉ) is the closure of the set ๐’Ÿ(ฮฉ) in ๐’ฎ(โ„2) with respect to the topology of ๐’ฎ(โ„2) [5], ยง1 in Section 5.
๐ป๐‘ (ฮฉ) is a Banach space with respect to the norm โ€–โ‹…โ€–๐ป๐‘ (ฮฉ) given byโ€–๐‘“โ€–๐ป๐‘ (ฮฉ)๎€ฝ=infโ€–๐นโ€–๐ป๐‘ (โ„2)โˆฃ๐นโˆˆ๐ป๐‘ ๎€ทโ„2๎€ธ๎€พ.isacontinuationof๐‘“(3.26) The magnetic boundary fields in part (c) of Theorem 3.2 corresponding to global finite energy solutions may also be reinterpreted as distributions restricted to ฮฉ. In this case we obtain ๐‘1,0,๐‘2,0โŽ›โŽœโŽœโŽ,โˆ‡โ‹…โˆ’๐‘2,0๐‘1,0โŽžโŽŸโŽŸโŽ โˆˆ๐ปโˆ’1/2(ฮฉ).(3.27) In general the conditions (3.27) are weaker than the conditions (3.18) in Theorem 3.2, and we assume that they are fulfilled for diffraction solutions with local finite energy.
The conditions (3.27), where ๐‘1,0 and ๐‘2,0 are considered only in the aperture ฮฉ, reflects the physical fact that ๐‘1,0 and ๐‘2,0 are only prescribed in ฮฉ. Namely, ๐‘1,0 and ๐‘2,0 are the tangential magnetic components of the incoming electromagnetic wave in the aperture ฮฉ.

4. The Local Energy Condition

In this section we derive the following characterisation for the diffraction solutions of local finite energy.

Theorem 4.1. Let be ๐‘’1,0,๐‘’2,0โˆˆ๐’ฎ๎…ž(โ„2) and supp๐‘’1,0,supp๐‘’2,0โŠ‚ฮฉ. Let ๐‘’๐‘—,๐‘๐‘—โˆถโ„‹โ†’โ„‚, ๐‘—=1,2,3, be defined as in Theorem 2.1. Then the diffraction solution ๐‘’๐‘—,๐‘๐‘—, ๐‘—=1,2,3, has local finite energy if and only if ๐‘’1,0,๐‘’2,0โŽ›โŽœโŽœโŽ,โˆ‡โ‹…โˆ’๐‘’2,0๐‘’1,0โŽžโŽŸโŽŸโŽ โˆˆ๎‚๐ปโˆ’1/2(ฮฉ).(4.1)

For proving Theorem 4.1 firstly we formulate the subsequent Lemma 4.2. Then, using this lemma, we give the proof of the theorem. Afterwards we prove the lemma.

In the sequel we will make use of the following notations. For ๐‘Ÿ>0 we define the open ball ๐ต๐‘Ÿ by๐ต๐‘Ÿ๎€ฝโˆถ=๐‘ฅโˆˆโ„2๎€พ.โˆฃ|๐‘ฅ|<๐‘Ÿ(4.2) For ๐‘Ÿ>0 and โ„Ž>0 the open cylinder ๐‘๐‘Ÿ,โ„Ž is defined by๐‘๐‘Ÿ,โ„Ž๎€ฝ๐‘ฅโˆถ==๎€ท๐‘ฅ,๐‘ฅ3๎€ธโˆˆโ„3โˆฃ๐‘ฅโˆˆ๐ต๐‘Ÿ,0<๐‘ฅ3๎€พ.<โ„Ž(4.3)

Lemma 4.2. Let be ๐‘’1,0,๐‘’2,0โˆˆ๐’ฎโ€ฒ(โ„2) and supp๐‘’1,0,supp๐‘’2,0โŠ‚ฮฉ. Let ๐‘’๐‘—,๐‘๐‘—โˆถโ„‹โ†’โ„‚, ๐‘—=1,2,3, be the diffraction solution given in Theorem 2.1. Let be ๐‘…โ€ฒ>0 with ฮฉโŠ‚๐ต๐‘…๎…ž and let be ๐‘…>๐‘…โ€ฒ and ๐ป>0. Then one has the following equivalences and implications.
(a)๐‘’๐‘—โˆˆ๐ฟ2(๐‘๐‘…,๐ป)โ‡”๐‘’๐‘—โˆˆ๐ฟ2(โ„2ร—(0,๐ป))for๐‘—โˆˆ{1,2}. (b)๐‘3โˆˆ๐ฟ2(๐‘๐‘…,๐ป)โ‡”๐‘3โˆˆ๐ฟ2(โ„2ร—(0,๐ป)). (c)๐‘’๐‘—โˆˆ๐ฟ2(โ„2ร—(0,๐ป))โ‡”๐‘’๐‘—,0โˆˆ๎‚๐ปโˆ’1/2(ฮฉ)for๐‘—โˆˆ{1,2}. (d)๐‘3โˆˆ๐ฟ2(โ„2ร—(0,๐ป))โ‡”๐‘3,0โˆˆ๎‚๐ปโˆ’1/2(ฮฉ). (e)๐‘’1,0,๐‘’2,0โˆˆ๎‚๐ปโˆ’1/2(ฮฉ)โ‡’๐‘’3โˆˆ๐ฟ2(๐‘๐‘…,๐ป). (f)๐‘’1,0,๐‘’2,0๎€ท,โˆ‡โ‹…โˆ’๐‘’2,0๐‘’1,0๎€ธโˆˆ๎‚๐ปโˆ’1/2(ฮฉ)โ‡’๐‘1,๐‘2โˆˆ๐ฟ2(๐‘๐‘…,๐ป).

Proof of Theorem 4.1. Let there be given ๐‘’1,0,๐‘’2,0โˆˆ๐’ฎ๎…ž(โ„2) with support in ฮฉand let ๐‘’๐‘—,๐‘๐‘—, ๐‘—=1,2,3, be defined as in Theorem 2.1.
Firstly, we assume that the diffraction solution ๐‘’๐‘—,๐‘๐‘—, ๐‘—=1,2,3, has local finite energy and show the validity of (4.1). Since the diffraction solution has local finite energy especially we have ๐‘’1,๐‘’2,๐‘3โˆˆ๐ฟ2(๐‘๐‘…,๐ป). From the parts (a) and (b) of Lemma 4.2 we find that ๐‘’1,๐‘’2,๐‘3โˆˆ๐ฟ2(โ„2ร—(0,๐ป)). From the parts (c) and (d) of the lemma we thus obtain ๐‘’1,0,๐‘’2,0,๐‘3,0โˆˆ๎‚๐ปโˆ’1/2(ฮฉ). Now the validity of (4.1) follows from๐‘3,0=๐‘–๐‘˜โŽ›โŽœโŽœโŽโˆ‡โ‹…โˆ’๐‘’2,0๐‘’1,0โŽžโŽŸโŽŸโŽ (4.4) (cf. (3.9)).
For proving the other direction, we assume that relation (4.1) is valid. Because of (4.4) from the parts (c) and (d) of Lemma 4.2 it follows that ๐‘’1,๐‘’2,๐‘3โˆˆ๐ฟ2(โ„2ร—(0,๐ป)). Regarding the other three field components, from the parts (e) and (f) of the lemma we obtain ๐‘’3,๐‘1,๐‘2โˆˆ๐ฟ2(๐‘๐‘…,๐ป). Since ๐‘…>๐‘…โ€ฒ and ๐ป>0 can be chosen arbitrarily large, the diffraction solution ๐‘’๐‘—,๐‘๐‘—, ๐‘—=1,2,3, has local finite energy.

We have yet to prove Lemma 4.2.

Proof of Lemma 4.2. Let the assumptions of Lemma 4.2 be fulfilled.
Proof of Part (a)
Let be ๐‘—โˆˆ{1,2}. Obviously we have only to show the validity of the implication๐‘’๐‘—โˆˆ๐ฟ2๎€ท๐‘๐‘…,๐ป๎€ธโŸน๐‘’๐‘—โˆˆ๐ฟ2๎€ทโ„2๎€ธ.ร—(0,๐ป)(4.5)
We set ๐œ€=(๐‘…โˆ’๐‘…โ€ฒ)/2 and ๐‘…๎…ž๎…ž=๐‘…โ€ฒ+๐œ€. For ๐œ‘โˆˆ[0,2๐œ‹) we consider the rotation matrix ๐ด๐œ‘=โŽ›โŽœโŽœโŽโŽžโŽŸโŽŸโŽ cos๐œ‘โˆ’sin๐œ‘sin๐œ‘cos๐œ‘(4.6) and define the function ๐‘’๐‘—(๐œ‘)โˆถโ„‹โ†’โ„‚by ๐‘’๐‘—(๐œ‘)(๐‘ฅ,๐‘ฅ3)=๐‘’๐‘—(๐ด๐œ‘๐‘ฅ,๐‘ฅ3)for ๐‘ฅโˆˆโ„2 and ๐‘ฅ3>0. We show that ๐‘’๐‘—(๐œ‘)โˆˆ๐ฟ2๐‘…๎€ท๎€ท๎…ž๎…ž๎€ธ๎€ธ[,โˆžร—โ„ร—(0,๐ป)for๐œ‘โˆˆ0,2๐œ‹),(4.7) regardless of whether ๐‘’๐‘—โˆˆ๐ฟ2(๐‘๐‘…,๐ป) or not.
From (4.7) we have ๐‘’๐‘—โˆˆ๐ฟ2๎‚€๐ด๐œ‘๐‘…๎‚€๎‚€โ€ฒ๎…ž๎‚๎‚๎‚[,โˆžร—โ„ร—(0,๐ป)for๐œ‘โˆˆ0,2๐œ‹),(4.8) where ๐ด๐œ‘((๐‘…โ€ฒ๎…ž,โˆž)ร—โ„) is the image of the set (๐‘…๎…ž๎…ž,โˆž)ร—โ„under the linear mapping ๐ด๐œ‘, that is, the set (๐‘…๎…ž๎…ž,โˆž)ร—โ„rotated by the angle ๐œ‘.
Since ๐‘…๎…ž๎…ž<๐‘…, by considering a sufficient number of angles ๐œ‘, from (4.8) we find that there is a polygon ๐‘ƒโŠ‚๐ต๐‘… with ๐‘’๐‘—โˆˆ๐ฟ2((โ„2โงต๐‘ƒ)ร—(0,๐ป)). Thus ๐‘’๐‘—โˆˆ๐ฟ2โ„๎€ท๎€ท2๎€ธร—(0,๐ป)โงต๐‘๐‘…,๐ป๎€ธ,(4.9) and therefore (4.5) holds true.
It remains to prove (4.7). To this end we use a contour integration technique which one of the authors had already used earlier in the treatment of diffraction problems [6, 7].
From the representation of ๐‘’๐‘— given in Theorem 2.1 we conclude that ๐‘’๐‘—(๐œ‘)๎€ท๐‘ฅ1,๐‘ฅ2,๐‘ฅ3๎€ธ=1๎€2๐œ‹โ„๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๐‘‘๐œ‰2,๎€ท๐‘ฅ1,๐‘ฅ2,๐‘ฅ3๎€ธโˆˆโ„‹,(4.10) where the function ๐›พ๐œ‘ is given by ๐›พ๐œ‘(๐œ‰)=ฬ‚๐‘’๐‘—,0๎€ท๐ด๐œ‘๐œ‰๎€ธ.(4.11) By assumption, it holds that supp๐‘’๐‘—,0โŠ‚ฮฉโŠ‚๐ต๐‘…โ€ฒ. Thus, from the Paley-Wiener theorem and the fact that ๐ด๐œ‘ is an orthogonal matrix it follows that ||๐›พ๐œ‘||๎‚€||๐œ‰||(๐œ‰)โ‰ค๐‘1+2๎‚๐‘/2๐‘’๐‘…โ€ฒ|Im๐œ‰|,๐œ‰โˆˆโ„‚2,(4.12) for some constants ๐‘>0 and ๐‘โˆˆโ„•0.
We split the inner integral in (4.10) into one integral over the interval (โˆ’โˆž,0) and one over (0,โˆž). Firstly we consider the integral over (โˆ’โˆž,0).
For ๐‘Ÿ>0 we define the curve ๐œ‚๐‘Ÿ by ๐œ‚๐‘Ÿ=๎‚†๐‘Ÿ๐‘’๐‘–๐›ผโˆฃ๐œ‹2๎‚‡.โ‰ค๐›ผโ‰ค๐œ‹(4.13) Let ฮ“๐‘Ÿ be the closed contour consisting of the parts {๐‘–๐‘กโˆฃ0โ‰ค๐‘ก<๐‘Ÿ}, ๐œ‚๐‘Ÿ and (โˆ’๐‘Ÿ,0). By Cauchyโ€™s theorem we have ๎€œฮ“๐‘Ÿ๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1=0,(4.14) here for nonreal values of ๐œ‰1 the function ๐‘š is defined by analytic continuation of the function ๐œ‰1โ†ฆ๐‘š(๐œ‰1,๐œ‰2), where ๐œ‰1<0 and |๐œ‰|โ‰ ๐‘˜, and โŸจ๐‘ฅ,๐œ‰โŸฉ=๐‘ฅ1๐œ‰1+๐‘ฅ2๐œ‰2.
Because of (4.12), for ๐‘ฅ1>๐‘…โ€ฒ it holds that lim๐‘Ÿโ†’โˆž๎€œ๐œ‚๐‘Ÿ๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1=0.(4.15) From the last two equations we obtain ๎€œ0โˆ’โˆž๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๎€œ=โˆ’0๐‘–โˆž๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1,๐‘ฅ1>๐‘…โ€ฒ.(4.16)
Now we treat the integral over (0,โˆž). Let โˆšโˆ— be the principal branch of the square root function (branch cut (โˆ’โˆž,0)). For ๐‘ง<0 we set โˆš๐‘งโˆ—โˆš=โˆ’๐‘–โˆš|๐‘ง|,thatis,โˆ— is thought to be defined on the negative real half-axis by continuation from the lower complex half-plane. In what follows, the function ๐‘šโˆ— is defined by ๐‘šโˆ—๎”(๐œ‰)=๐‘˜2โˆ’๐œ‰21โˆ’๐œ‰22โˆ—. Regarding the integration variable ๐œ‰2 we distinguish the two cases |๐œ‰2|<๐‘˜ and |๐œ‰2|>๐‘˜.
Firstly we assume that |๐œ‰2|<๐‘˜. We have ๎€œโˆž0๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1=๎€œ๎”๐‘˜2โˆ’๐œ‰220๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘šโˆ—(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1+๎€œโˆž๎”๐‘˜2โˆ’๐œ‰22๐›พ๐œ‘(๐œ‰)๐‘’โˆ’๐‘–๐‘ฅ3๐‘šโˆ—(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1=๎€œโˆž0๐›พ๐œ‘(๐œ‰)๐‘’โˆ’๐‘–๐‘ฅ3๐‘šโˆ—(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๎€œ+2๐‘–๎”๐‘˜2โˆ’๐œ‰220๐›พ๐œ‘๎€ท๐‘ฅ(๐œ‰)sin3๐‘šโˆ—๎€ธ๐‘’(๐œ‰)๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1.(4.17) By contour integration analogous as in the derivation of (4.16), applied to the next-to-last integral, eventually we get ๎€œโˆž0๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1=๎€œ0๐‘–โˆž๐›พ๐œ‘(๐œ‰)๐‘’โˆ’๐‘–๐‘ฅ3๐‘šโˆ—(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๎€œ+2๐‘–๎”๐‘˜2โˆ’๐œ‰220๐›พ๐œ‘๎€ท๐‘ฅ(๐œ‰)sin3๐‘šโˆ—๎€ธ๐‘’(๐œ‰)๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1||๐œ‰for2||<๐‘˜,๐‘ฅ1>๐‘…๎…ž.(4.18)
In the case |๐œ‰2|>๐‘˜ it holds that ๐‘š(๐œ‰)=โˆ’๐‘šโˆ—(๐œ‰)for ๐œ‰1>0, and contour integration yields ๎€œโˆž0๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1=๎€œ0๐‘–โˆž๐›พ๐œ‘(๐œ‰)๐‘’โˆ’๐‘–๐‘ฅ3๐‘šโˆ—(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1||๐œ‰for2||>๐‘˜,๐‘ฅ1>๐‘…๎…ž.(4.19)
In the following, for ๐œ‰1โˆˆโ„or ๐œ‰1โˆˆ{๐‘–๐‘ก|0<๐‘ก<โˆž} we set ๎”๐‘š(๐œ‰)=๐‘˜2โˆ’๐œ‰21โˆ’๐œ‰22, where the square root is chosen in the way that Re๐‘š(๐œ‰)โ‰ฅ0and Im๐‘š(๐œ‰)โ‰ฅ0. This definition is in accordance with (2.15). In this notation, from (4.18) and (4.19) we obtain ๎€œโ„๎€œโˆž0๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๐‘‘๐œ‰2=๎€œ๐‘˜โˆ’๐‘˜๎€œ0๐‘–โˆž๐›พ๐œ‘(๐œ‰)๐‘’โˆ’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๐‘‘๐œ‰2๎€œ+2๐‘–๐‘˜โˆ’๐‘˜๎€œ๎”๐‘˜2โˆ’๐œ‰220๐›พ๐œ‘(๎€ท๐‘ฅ๐œ‰)sin3๎€ธ๐‘’๐‘š(๐œ‰)๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๐‘‘๐œ‰2+๎€œ[]โ„โงตโˆ’๐‘˜,๐‘˜๎€œ๐‘–๎”๐œ‰22โˆ’๐‘˜20๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๐‘‘๐œ‰2+๎€œ[]โ„โงตโˆ’๐‘˜,๐‘˜๎€œ๐‘–๎”๐‘–โˆž๐œ‰22โˆ’๐‘˜2๐›พ๐œ‘(๐œ‰)๐‘’โˆ’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๐‘‘๐œ‰2for๐‘ฅ1>๐‘…๎…ž.(4.20) Together with (4.16) we thus find ๎€โ„๐›พ๐œ‘(๐œ‰)๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)+๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๐‘‘๐œ‰2๎€œ=2๐‘–๐‘˜โˆ’๐‘˜๎€œ๎”๐‘˜2โˆ’๐œ‰220๐›พ๐œ‘๎€ท๐‘ฅ(๐œ‰)sin3๎€ธ๐‘’๐‘š(๐œ‰)๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๐‘‘๐œ‰2๎€œโˆ’2๐‘–๐‘˜โˆ’๐‘˜๎€œ0๐‘–โˆž๐›พ๐œ‘๎€ท๐‘ฅ(๐œ‰)sin3๎€ธ๐‘’๐‘š(๐œ‰)๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๐‘‘๐œ‰2๎€œโˆ’2๐‘–[]โ„โงตโˆ’๐‘˜,๐‘˜๎€œ๐‘–๎”๐‘–โˆž๐œ‰22โˆ’๐‘˜2๐›พ๐œ‘๎€ท๐‘ฅ(๐œ‰)sin3๎€ธ๐‘’๐‘š(๐œ‰)๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰1๐‘‘๐œ‰2for๐‘ฅ1>๐‘…๎…ž.(4.21)
Now we show that each of the three addends in the right-hand side of (4.21), considered as a function of ๐‘ฅ=(๐‘ฅ,๐‘ฅ3), lies in ๐ฟ2((๐‘…๎…ž๎…ž,โˆž)ร—โ„ร—(0,๐ป)). Then, because of the representation (4.10), the stated relation (4.7) is proved.
We begin with the first addend. Let ๐ดโˆถ={๐œ‰โˆˆโ„2||๐œ‰|โ‰ค๐‘˜,๐œ‰1โ‰ฅ0}, let ๐œ’๐ด be the characteristic function of the set ๐ด and let ฬŒโ€Œ denote the inverse Fourier transform. There is a constant ๐‘>0, which does not depend on ๐‘ฅ3โˆˆ(0,๐ป), such that โ€–โ€–๎€บ๐œ’๐ด๐›พ๐œ‘๎€ท๐‘ฅsin3๐‘š(โ‹…)๎€ธ๎€ปฬŒโ€Œโ€–โ€–๐ฟ2(โ„2)=โ€–โ€–๐œ’๐ด๐›พ๐œ‘๎€ท๐‘ฅsin3๎€ธโ€–โ€–๐‘š(โ‹…)๐ฟ2(โ„2)โ‰ค๐‘.(4.22) From this it follows that the first addend in the right-hand side of (4.21) is quadratically integrable over the set โ„2ร—(0,๐ป) and thus especially quadratically integrable over (๐‘…๎…ž๎…ž,โˆž)ร—โ„ร—(0,๐ป).
Now we come to the second addend. We define ๎‚๐‘… by ๎‚๐‘…=(๐‘…โ€ฒ+๐‘…๎…ž๎…ž)/2=๐‘…โ€ฒ+(๐œ€/2). From (4.12) we find that there is a constant ๐‘1>0 with ||๐›พ๐œ‘๎€ท๐‘–๐‘ก,๐œ‰2๎€ธ||โ‰ค๐‘1๐‘’๎‚๐‘…๐‘กfor๐œ‰2โˆˆ[]โˆ’๐‘˜,๐‘˜,๐‘กโ‰ฅ0.(4.23) Since for ๐œ‰2โˆˆ[โˆ’๐‘˜,๐‘˜] and ๐‘กโ‰ฅ0 the quantity ๐‘š(๐‘–๐‘ก,๐œ‰2) is a real number and therefore |sin(๐‘ฅ3๐‘š(๐‘–๐‘ก,๐œ‰2))|โ‰ค1 for ๐‘ฅ3โˆˆ(0,๐ป), we conclude that ||||๎€œโˆž0๐›พ๐œ‘๎€ท๐‘–๐‘ก,๐œ‰2๎€ธ๎€ท๐‘ฅsin3๐‘š๎€ท๐‘–๐‘ก,๐œ‰2๐‘’๎€ธ๎€ธโˆ’๐‘ก๐‘ฅ1||||๐‘‘๐‘กโ‰ค๐‘1๎€œโˆž0๐‘’(๎‚๐‘…โˆ’๐‘ฅ1)๐‘ก๐‘๐‘‘๐‘ก=1๐‘ฅ1โˆ’๎‚๐‘…for๐œ‰2โˆˆ[]โˆ’๐‘˜,๐‘˜,๐‘ฅ3โˆˆ(0,๐ป),๐‘ฅ1>๎‚๐‘….(4.24) Thus we have ๎€œ๐‘˜โˆ’๐‘˜||||๎€œโˆž0๐›พ๐œ‘๎€ท๐‘–๐‘ก,๐œ‰2๎€ธ๎€ท๐‘ฅsin3๐‘š๎€ท๐‘–๐‘ก,๐œ‰2๐‘’๎€ธ๎€ธโˆ’๐‘ก๐‘ฅ1||||๐‘‘๐‘ก2๐‘‘๐œ‰2โ‰ค2๐‘˜๐‘21๎‚€๐‘ฅ1โˆ’๎‚๐‘…๎‚2for๐‘ฅ3โˆˆ(0,๐ป),๐‘ฅ1>๎‚๐‘….(4.25) Because the inverse Fourier transform (here with regard to the variable ๐œ‰2) is isometric on ๐ฟ2, we see that ๎€œโ„||||๎€œ๐‘˜โˆ’๐‘˜๎€œโˆž0๐›พ๐œ‘๎€ท๐‘–๐‘ก,๐œ‰2๎€ธ๎€ท๐‘ฅsin3๐‘š๎€ท๐‘–๐‘ก,๐œ‰2๐‘’๎€ธ๎€ธโˆ’๐‘ก๐‘ฅ1๐‘‘๐‘ก๐‘’๐‘–๐œ‰2๐‘ฅ2๐‘‘๐œ‰2||||2๐‘‘๐‘ฅ2โ‰ค4๐œ‹๐‘˜๐‘21๎‚€๐‘ฅ1โˆ’๎‚๐‘…๎‚2for๐‘ฅ3โˆˆ(0,๐ป),๐‘ฅ1>๎‚๐‘….(4.26) It follows that ๎€œ๐ป0๎€œโˆž๐‘…โ€ฒโ€ฒ๎€œโ„||||๎€œ๐‘˜โˆ’๐‘˜๎€œโˆž0๐›พ๐œ‘๎€ท๐‘–๐‘ก,๐œ‰2๎€ธ๎€ท๐‘ฅsin3๐‘š๎€ท๐‘–๐‘ก,๐œ‰2๐‘’๎€ธ๎€ธโˆ’๐‘ก๐‘ฅ1๐‘‘๐‘ก๐‘’๐‘–๐œ‰2๐‘ฅ2๐‘‘๐œ‰2||||2๐‘‘๐‘ฅ2๐‘‘๐‘ฅ1๐‘‘๐‘ฅ3<โˆž.(4.27) The substitution ๐‘ก=โˆ’๐‘–๐œ‰1 in the innermost integral now shows that the second addend in the right-hand side of (4.21) indeed lies in ๐ฟ2((๐‘…๎…ž๎…ž,โˆž)ร—โ„ร—(0,๐ป)).
Regarding the third addend, from (4.12) we find |||||๎€œโˆž๎”๐œ‰22โˆ’๐‘˜2๐›พ๐œ‘๎€ท๐‘–๐‘ก,๐œ‰2๎€ธ๎€ท๐‘ฅsin3๐‘š๎€ท๐‘–๐‘ก,๐œ‰2๐‘’๎€ธ๎€ธโˆ’๐‘ก๐‘ฅ1|||||๎€œ๐‘‘๐‘กโ‰ค๐‘โˆž๎”๐œ‰22โˆ’๐‘˜2๎€ท1+๐‘ก2+๐œ‰22๎€ธ๐‘/2๐‘’(๐‘…โ€ฒโˆ’๐‘ฅ1)๐‘ก๐‘‘๐‘กfor๐œ‰2[]โˆˆโ„โงตโˆ’๐‘˜,๐‘˜,๐‘ฅ3โˆˆ(0,๐ป),๐‘ฅ1>๐‘…๎…ž.(4.28) Now let be ๎‚๐‘…=๐‘…โ€ฒ+(๐œ€/2) as above. By reason of ๎€ท1+๐‘ก2+๐œ‰22๎€ธ๐‘/2โ‰ค๎€ท1+๐‘ก2๎€ธ๐‘/2๎€ท1+๐œ‰22๎€ธ๐‘/2,(4.29) we conclude that there are constants ๐‘2,๐‘3>0such that |||||๎€œโˆž๎”๐œ‰22โˆ’๐‘˜2๐›พ๐œ‘๎€ท๐‘–๐‘ก,๐œ‰2๎€ธ๎€ท๐‘ฅsin3๐‘š๎€ท๐‘–๐‘ก,๐œ‰2๐‘’๎€ธ๎€ธโˆ’๐‘ก๐‘ฅ1|||||๐‘‘๐‘กโ‰ค๐‘2๎€ท1+๐œ‰22๎€ธ๐‘/2๎€œโˆž๎”๐œ‰22โˆ’๐‘˜2๐‘’(๎‚๐‘…โˆ’๐‘ฅ1)๐‘ก=๐‘๐‘‘๐‘ก2๐‘ฅ1โˆ’๎‚๐‘…๎€ท1+๐œ‰22๎€ธ๐‘/2๐‘’(๎‚๐‘…โˆ’๐‘ฅ1)๎”๐œ‰22โˆ’๐‘˜2โ‰ค๐‘3๐‘ฅ1โˆ’๎‚๐‘…๐‘’๎”โˆ’(๐œ€/4)๐œ‰22โˆ’๐‘˜2for๐œ‰2[]โˆˆโ„โงตโˆ’๐‘˜,๐‘˜,๐‘ฅ3โˆˆ(0,๐ป),๐‘ฅ1>๐‘…โ€ฒ๎…ž.(4.30) This estimate corresponds to formula (4.24), used in the case of the second addend. Continuing as in this latter case, it is seen that the third addend in the right-hand side of (4.21) lies in ๐ฟ2((๐‘…๎…ž๎…ž,โˆž)ร—โ„ร—(0,๐ป)) too.
Proof of Part (b)
The preceding proof of part (a) is based on the fact that for ๐‘—โˆˆ{1,2} it holds that ฬ‚๐‘’๐‘—๎€ท๐œ‰,๐‘ฅ3๎€ธ=๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)ฬ‚๐‘’๐‘—,0(๐œ‰)for๐œ‰โˆˆโ„2,๐‘ฅ3>0,(4.31)supp๐‘’๐‘—,0โŠ‚ฮฉ.(4.32) The relation (4.31), which is equivalent to the representation of ๐‘’๐‘— given in Theorem 2.1, has led to (4.10).
Since the third magnetic component ๐‘3 fulfills conditions which are analogous to (4.31) and (4.32), the proof of part (b) of the lemma is completely analogous to the proof of part (a). The condition for ๐‘3 that is analogous to (4.31) is given in (2.19); note that (2.19) also holds for |๐œ‰|<๐‘˜. The condition supp๐‘3,0โŠ‚ฮฉis a direct consequence of ๐‘3,0=๐‘–๐‘˜โŽ›โŽœโŽœโŽโˆ‡โ‹…โˆ’๐‘’2,0๐‘’1,0โŽžโŽŸโŽŸโŽ ,(4.33) following from the second equation in (2.18), and supp๐‘’1,0,supp๐‘’2,0โŠ‚ฮฉ.
Proof of Parts (c) and (d)
These parts of the lemma can be proved along the lines of the proof of Theorem 3.1. For example, with respect to ๐‘’๐‘—, ๐‘—โˆˆ{1,2}, one has ๎€œ๐ป0๎€โ„2||๐‘’๐‘—๎€ท๐‘ฅ,๐‘ฅ3๎€ธ||2๐‘‘๐‘ฅ๐‘‘๐‘ฅ3=๎€œ๐ป0๎€โ„2||ฬ‚๐‘’๐‘—๎€ท๐œ‰,๐‘ฅ3๎€ธ||2๐‘‘๐œ‰๐‘‘๐‘ฅ3=๎€โ„2||ฬ‚๐‘’๐‘—,0||(๐œ‰)2๎€œ๐ป0||๐‘’๐‘–๐‘ฅ3๐‘š(๐œ‰)||2๐‘‘๐‘ฅ3๎€๐‘‘๐œ‰=๐ป||๐œ‰||<๐‘˜||ฬ‚๐‘’๐‘—,0||(๐œ‰)2๎€๐‘‘๐œ‰+||๐œ‰||>๐‘˜||ฬ‚๐‘’๐‘—,0||(๐œ‰)21โˆ’๐‘’โˆšโˆ’2๐ป|๐œ‰|2โˆ’๐‘˜22๎”||๐œ‰||2โˆ’๐‘˜2๐‘‘๐œ‰.(4.34) The condition (3.12), used in the proof of Theorem 3.1 because the factor ๐‘š(๐œ‰)โˆ’1 is not locally square integrable, is not needed for the special field components ๐‘’1, ๐‘’2, and ๐‘3.
Proof of Part (e)
From the first equation in (2.18) and the first equation in (2.19) we obtainฬ‚๐‘’3๎€ท๐œ‰,๐‘ฅ3๎€ธ1=โˆ’๐‘š๎€บ๐œ‰(๐œ‰)1ฬ‚๐‘’1๎€ท๐œ‰,๐‘ฅ3๎€ธ+๐œ‰2ฬ‚๐‘’2๎€ท๐œ‰,๐‘ฅ3๎€ธ๎€ป,๐œ‰โˆˆโ„2,๐‘ฅ3โ‰ฅ0.(4.35) Thus for (๐‘ฅ,๐‘ฅ3)โˆˆโ„‹ we have ๐‘’3๎€ท๐‘ฅ,๐‘ฅ3๎€ธ1=โˆ’๎€2๐œ‹โ„21๎€บ๐œ‰๐‘š(๐œ‰)1ฬ‚๐‘’1๎€ท๐œ‰,๐‘ฅ3๎€ธ+๐œ‰2ฬ‚๐‘’2๎€ท๐œ‰,๐‘ฅ3๐‘’๎€ธ๎€ป๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰.(4.36) Now we define the function ๐‘š0โˆถโ„2โ†’โ„‚by ๐‘š0(๎”๐œ‰)=๐‘–๐‘˜2+||๐œ‰||2(4.37) and the function ๐‘’3(0)โˆถโ„‹โ†’โ„‚by ๐‘’3(0)๎€ท๐‘ฅ,๐‘ฅ3๎€ธ1=โˆ’๎€2๐œ‹โ„21๐‘š0๎€บ๐œ‰(๐œ‰)1ฬ‚๐‘’1๎€ท๐œ‰,๐‘ฅ3๎€ธ+๐œ‰2ฬ‚๐‘’2๎€ท๐œ‰,๐‘ฅ3๐‘’๎€ธ๎€ป๐‘–โŸจ๐‘ฅ,๐œ‰โŸฉ๐‘‘๐œ‰.(4.38) Using (4.36), we find ||๐‘’3(0)๎€ท๐‘ฅ,๐‘ฅ3๎€ธโˆ’๐‘’3๎€ท๐‘ฅ,๐‘ฅ3๎€ธ||โ‰ค12๐œ‹2๎“๐‘—=1๎‚€๐ผ๐‘—(1)๎€ท๐‘ฅ3๎€ธ+๐ผ๐‘—(2)๎€ท๐‘ฅ3๎€ธ๎‚,(4.39) with ๐ผ๐‘—(1)๎€ท๐‘ฅ3๎€ธ=๎€||๐œ‰||<2๐‘˜||||1๐‘š0โˆ’1(๐œ‰)||||||๐œ‰๐‘š(๐œ‰)๐‘—||||ฬ‚๐‘’๐‘—๎€ท๐œ‰,๐‘ฅ3๎€ธ||๐ผ๐‘‘๐œ‰,๐‘—=1,2,๐‘—(2)๎€ท๐‘ฅ3๎€ธ=๎€|๐œ‰|>2๐‘˜||||1๐‘š0โˆ’1(๐œ‰)||||||๐œ‰๐‘š(๐œ‰)๐‘—||||ฬ‚๐‘’๐‘—๎€ท๐œ‰,๐‘ฅ3๎€ธ||๐‘‘๐œ‰,๐‘—=1,2.(4.40) From (4.31) we see that there is a constant ๐‘1>0 such that ||๐ผ๐‘—(1)๎€ท๐‘ฅ3๎€ธ||โ‰ค๐‘1for๐‘—โˆˆ{1,2},๐‘ฅ3>0.(4.41) For |๐œ‰|>2๐‘˜ it holds that ||||1๐‘š0โˆ’1(๐œ‰)||||๎ƒฉ1๐‘š(๐œ‰)=๐‘‚||๐œ‰||3๎ƒช.(4.42) Therefore with some constant ๐‘2>0 we have for ๐‘—โˆˆ{1,2} and ๐‘ฅ3>0||๐ผ๐‘—(2)๎€ท๐‘ฅ3๎€ธ||โ‰ค๐‘2๎€||๐œ‰||>2๐‘˜1||๐œ‰||3/2|||||1||๐œ‰||1/2ฬ‚๐‘’๐‘—๎€ท๐œ‰,๐‘ฅ3๎€ธ|||||๐‘‘๐œ‰.(4.43) From this we get from the Cauchy-Schwarz inequality that ||๐ผ๐‘—(2)๎€ท๐‘ฅ3๎€ธ||โ‰ค๐‘2๎ƒฉ๎€||๐œ‰||>2๐‘˜1||๐œ‰||3๎ƒช๐‘‘๐œ‰1/2๎ƒฉ๎€||๐œ‰||>2๐‘˜1||๐œ‰||||ฬ‚๐‘’๐‘—(๐œ‰,๐‘ฅ3)|