spectral analysis of a dissipative problem in electrodynamics: the...

Acta Applicandae Mathematicae 6 (1986), 63-94. 63 (~) 1986 by D. Reidel Publishing Company.

Spectral Analysis of a Dissipative Problem in Electrodynamics: The Sommerfeld Problem

D A V I D S. G I L L I A M Texas Tech. University, Lubbock, TX 79409, U.S.A.

and

JOHN R. SCHULENBERGER ANAH Corp., Tucson, A Z 85719, U.S.A.

(Received: 8 January 1985; revised: 20 June 1985)

Abstract. This paper contains a survey of Gilliam and Schulenberger's work entitled 'The propagation of electromagnetic waves through, along and over a three dimensional conducting half space'. A special case of this problem was originally considered in the now classic paper by A. Sommerfeld in 1909. Since that time there have been a tremendous number of articles in engineering and physics literature on special cases of this important physical model. However, the real composition of the electromagnetic field due to general finite energy sources remained unclear. In particular, there has been considerable controversy regarding the existence and properties of the Zenneck surface wave.

The extreme length and technical nature of the complete solution found in Gilliam and Schulen- berger's work as well as the rich history and physical importance of this problem provide the motivation for this survey. Furthermore, the techniques used in the work provide a basis for the rigorous mathematical analysis of a great many dissipative problems in electrodynamics within a unified mathematical framework.

AMS (MOS) subject classifications (1980). 35F10, 35L45, 35P10, 78A40.

Key words. Maximal dissipative operator, generalized eigenfunction expansion, representations of contraction semigroups, electromagnetic wave propagation.

O. I n t r o d u c t i o n a n d H i s t o r i c a l D i s c u s s i o n

T h e p r e sen t p r o b l e m arose in the ear l ies t days of rad io in the a t t e m p t to

u n d e r s t a n d how rad io waves p r o p a g a t e f rom a t r ansmi t t e r to a r e c e i v e r while

o v e r c o m i n g the ef fec t of the E a r t h ' s c u r v a t u r e ( the ro le of the i o n o s p h e r e was

no t k n o w n at tha t t ime). In 1907, Z e n n e c k [32] found tha t Maxwe l l ' s equa t ions

for t i m e - h a r n o m i c , t r ansve r se m a g n e t i c (TM) waves in two space d imens ions

have solut ions which d e c a y exponen t i a l l y away f rom the in te r face b e t w e e n a

c o n d u c t i n g and n o n c o n d u c t i n g half space and are i n t e rp re t ed as sur face waves .

In 1909, S o m m e r f e l d [24] m a d e an ex tens ive s tudy of the s t a t iona ry field due to a

ve r t i c a l d ipole . B e c a u s e of cy l indr ica l symmet ry , this p r o b l e m r e duc e s to a

p r o b l e m for the sca la r H e l m h o l t z e q u a t i o n in two i n d e p e n d e n t var iab les . Som-

m e r f e ld o b t a i n e d a r e p r e s e n t a t i o n of the so lu t ion which, when wr i t t en in a

64 DAVID S. GILLIAM AND JOHN R. SCHULENBERGER

particular way, displayed a component which he identified with Zenneck's surface wave. In 1919, Weyl [31] obtained a different representation of this solution. Weyl's solution decomposes asymptotically into spatial-wave and surface-wave components which are not the same as those obtained by Sommerfeld. Sub- sequently, many other types of surface-wave components were found, but the real composition of the solution remained unclear. Good discussions of this problem and its history can be found in a variety of sources (see, e.g., [1, 4, 7, 8, 16, 26, 27]).

Another approach to this problem is to eliminate the conducting half space by imposing a dissipative (Leontovich) boundary condition at the interface. The time-harmonic, steady-state radiation due to a vertical dipole is studied in this way in [8], while a representation of the solution to the general initial boundary- value problem in three dimensions is given in [10]. It will become apparent below that this approximation of the general problem is much less felicitous than commonly believed.

There is an enormous amount of literature on special cases of this problem, and a great deal has been gained from this work. In spite of this, a complete understanding of the composition of the solutions for general types of sources remained unanswered. In part, this is because the composition of the solution in terms of TM (transverse magnetic) and TE (transverse electric) plane waves, surface waves, etc., depends, in general, on the particular initial source. Tradi- tionally, wave-propagation problems are studied for very specific sources and usually under the assumption that the source is harmonic in time (for example, the steady-state response to a harmonically-radiating vertical or horizontal dipole antenna).

Traditional approaches to the analysis of these problems via dyadic Green's functions or introduction of a Hertz potential and reduction to a problem for the scalar wave equation require the introduction of an Ansatz; the interpretation of the solution obtained is thus Ansatz-dependent. These techniques generate solutions of the equations, but it is not clear that the complete solution to a well-posed problem is obtained in this way. Questions such as this account for a substantial part of the rich history of functional analysis in which the first rigorous proofs can be found of the completeness of the eigenfunction expansions for the classical special functions of mathematical physics. In general, proofs of completeness for general classes of initial data (sources) are extremely difficult.

The development of the type of machinery needed to carry out a rigorous and complete analysis of such problems for general finite energy sources, even in the simplest cases of nonconducting media, required many years of effort by some of the most noted mathematicians in recent history [2, 6, 25], e.g., Riesz, von Neumann, Stone, and Weyl. Their efforts resulted in the development of the spectral theory of unbounded self-adjoint operators in Hilbert space. It was this theory which finally provided the rigorous proofs of completeness of the eigenfunction expansions obtained in the Sturm-LiouviUe boundary value problems

THE SOMMERFELD PROBLEM 65

which form the basis of the most common approach to analyzing electromagnetic problems in engineering literature. Unfortunately, the spectral theory of self- adjoint operators does not extend in a simple way to dissipative problems of electrodynamics. The development of such a theory requires, at least initially, the careful analysis of specific examples with the hope of eventually formulating and proving general theorems. On the brighter side, most of the 'simpler' models (such as the one considered here) are already important physically and are, thus, worth the individual effort.

Over the past several years, Schulenberger and Gilliam [9-13, 19-21] have developed special techniques (based on generalized eigenfunction expansions) for treating problems of wave propagation in layered media in the presence of dissipation. A major advantage of this analysis is that the solution is obtained in a physically and computationally useful form for all finite energy initial data (sources). Beyond this, it is now possible to incorporate a great many wave- propagation problems of classical physics within a unified mathematical framework. The procedure consists in constructing the resolvent for the appropriate spatial Maxwell operator, integrating the resolvent around the spectrum in two different ways and, finally, passing to the limit onto the spectrum to obtain the eigenfunction expansion and Parseval identity (the completeness of the eigenfunction expansion).

We shall now describe the general problem in more detail. Electromagnetic wave propagation in layered media for Maxwell's equations can be formulated as an initial boundary value problem for an evolution equation in an infinite- dimensional energy Hilbert space H in the form

i O~u(x, t) = A(x, D)u(x, t) (0.1)

where A is the spatial Maxwell operator (see (2.2) below), D = (D1,/)2, D3),

D i = - i 0 i (0 r is the partial derivative with respect to x~), and u(x, t) = t(ul, u 2) u a = t(Ei, E2, E3), u 2 = '(H1, //2, /43) are, respectively, the electric and magnetic field intensities at a point x in R 3 at time t. The problem is to find u(x, 0 subject to a prescribed initial condition at time t = 0. In order to obtain a representation of the solution for all finite energy initial data one must first define the operator A in some 'maximal' sense in the Hilbert space H. This guarantees the existence and uniqueness of solutions for initial data in the domain of A. Following the notions of abstract semigroup theory [14], one then obtains a semigroup (with generator A) S(t)= exp(-iAt) of operators in H which delivers the solution to the initial value problem; namely, u(x, t)= S(t)f(x), where u(x, O)= f(x) is the prescribed initial condition.

There has been a fair amount of progress regarding the uniqueness and existence of solutions to some general problems of this type in the theory of semigroups and maximal dissipative operators in Hilbert space. Related transient and steady-state analysis of problems of this type can be found in [22, 23, 28-30]. As was mentioned earlier, there is also a vast engineering literature available on


such problems [1, 3, 5, 7, 26, 27] from which substantial information has been obtained for a great many specific types of sources.

As we have pointed out previously [12], part of the confusion regarding surface waves is that in a formal Ansatz approach adapted to special sources or initial data the interpretation of the eventual solution is Ansatz-dependent: in one form the solution may seem to have components with certain properties, while these properties or even various components themselves may not be evident in another form [11]. In the Sommerfeld problem the various types of waves from which the solution u(x, t) = S( t ) f (x) is synthesized are completely specified by the spectrum of the generator of S(t). Once this is known, there can be no confusion whatever regarding the composition of the solution. Our results explicitly and unequivocally describe the structure of the solution for any finite-energy initial source or initial field configuration.

Our approach to obtaining a representation of S(t) = exp(-iAt) is essentially to resurrect the ideas that led to the development of spectral theory as it applies to complete eigenfunction expansions. The present problem is not 'spectral', i.e., the generator of S(t) does not have a bounded spectral family, so we can expect no assistance from available theory. Now the principal contribution of spectral theory in those cases where it is applicable is the Parseval identity, i.e., the representation of an arbitrary initial datum in terms of the (generalized) eigenfunctions of the generator A of S(t). The basis for this in the case of a self-adjoint operator A with resolvent GA(~) = ( A - ~I) -1, ~ ~ C, is the familiar formula

I = (2 ~-i)-llim [ [GA(A + iK) - GA(;t - iK)] d;t. (0.2) ~¢-"~0 JA

In order to interpret this formula in a manner that generalizes to nonself-adjoint operators, we first observe that if A is a bounded operator (0.2) states that the integral of the resolvent around the spectrum is the identity; this is simply Cauchy's theorem in the Branch algebra of bounded operators. For unbounded operators A we can also interpret (0.2) as an integral around the spectrum in the sense that

I = (27ri) -1 lim lim [ GA(~) dff (0.3) K---*0 N---,o~ 2CN(K)

where CN(K) = C~(K) U C~(K), C~(K) = {~ = ;~ + iK, - N < )t < N}, and C~(K) = {~ = ;t -- iK, N > ,~ ~ - N } is the oriented line from N - iK to - N - iK. In the form (0.3) formula (0.2) generalizes to the nonself-adjoint operator considered here (see also [10]).

Our program is thus to find the spectrum of A, obtain a representation of the resolvent G(~)= (A-~i) -1 , and integrate the latter around the spectrum. We then pass to the same limits as in (0.3) in two different ways: the first yields essentially the identity, just as in (0.3), while the second provides a resolution of the identity in terms of the generalized eigenfunctions of A and its adjoint A*.


The content of the paper is thus a systematic realization of this basic and very old idea.

From a practical point of view it is important to extend these results to the problem of a time dependent source term, i.e.,

i Otu(x, t ) - Au(x, t) = F(x, t), u(x, 0) = 0 (0.4)

where F(x, t) is a prescribed source term. For example, F(x, t) may represent a horizontal or vertical TM or TE dipole antenna. If t---> F(., t) c H is continuous, say, then the solution to (0.2) can be written as

Io' u(x, t) = S ( t - r)F(x, ~') d~'. (0.5)

In common physical models, however, the sources are taken to be singular and time harmonic, e.g., radiating dipoles, and one seeks the steady-state response. Mathematically, this amounts to a specialization of the 'outgoing' Green function G(x, y; o2) for the operator A where co ~ R is the frequency. The Green function is simply the limit of the resolvent kernel on the real part of the spectrum of A at (.o:

G(x, y; ¢0) = lira G(x, y; ~'). (0.6) ;;--->~o

Taking F(x, t)= exp(icot)f(x) in (0.4) we seek u(x, t)= exp(io)t)v(x) where v(x) satisfies

( A - ¢al)v(x) = f(x). (0.7)

For example, the steady-state response v(x) to a dipole source of frequency co in direction a located at the origin is simply

v(x) = G(x, 0; w)'(a~o, 0, 0, 0). (0.8)

For information regarding this approach see [16, 17, 19-21]. We now discuss the contents of the paper. We first give the mathematical

formulation of the initial value problem, define the dissipative Maxwell operator A in the energy space H, and describe various properties of functions in the domain of A. The condition that a function be in the domain of A implies that the tangential components of E and H are continuous across {xs = 0} (the interface condition). This fact is used later in the construction of the resolvents for A and its adjoint A*. We then determine dense sets of smooth, compactly supported functions in the eigensubspaces of A corresponding to the embedded eigenvalues 0 and - - iETlo r and derive the Gillberger decomposition of the space H. It is rather unexpected that if o" ¢ 0 a general function in the domain of A and orthogonal in H to the eigensubspaces need not have tangential derivatives in LZ; it does, however, if or = 0.

The next step in the analysis found in [11] is the explicit construction of the


resolvent kernels for A and its adjoint A*. At first glance this seems to require the computation of eight 6 x 6 reflection and transmission matrices from four 4 x 6 linear algebraic systems obtained from the aforementioned condition on the tangential components of the electric and magnetic fields. However, the so-called paramutation relations [9-12] suggest a form of writing these systems so that it is only necessary to solve two simple 2 × 2 systems to completely determine both resolvent kernels. We remark that this simplification persists for problems with several plane boundaries, e.g., asymmetric waveguides [9].

Using the resolvent kernels obtained above, we then determine the spectrum ,r(A) of A;

o'(A) = or(Ao) L,.J or(A1) L,.J {s±} L,J {So}.

Here o'(A0) and o'(A1) consist of the spectra of the operators engendered by Ao and A1 on all of R 3 (see (2.2) below); the points 0, -i*11cr are embedded eigenvalues, while the remaining points are points of the continuous spectrum corresponding to plane-wave modes of the two media filling R g and R 3. The sets {s±} form two curves in the complex plane each point of which corresponds to a surface mode having frequency s±(0, ~ ~ R 2. Finally, the set {So} corresponds to an interval of the imaginary axis, and to each point there corresponds an AH mode; the frequencies of AH modes have no real parts, so such modes do not propagate but rather simply decay in time. As far as we know, such modes have not been discovered previously. Superpositions of these modes provide initial data giving rise to pure AH-waves. We do not know if any such waves have ever been recorded physicaly nor what their significance might be.

The Parseval identity for A, A* expresses a function as a superposition of the generalized eigenfunctions of A, A* which takes the form of a sum of projections applied to the given function. Each projection consists of superpositions of generalized eigenfunctions of A, A* corresponding to the various types of modes mentioned. As indicated above, it is the heart of this work to establish this identity, for it is in this way that the general initial condition is satisfied. The extreme length and technical nature of the derivation of the Parseval identity provide the motivation for this survey. The situation here is more involved than in [10], but the idea is basically the same. Here, however, we meet a situation peculiar to dissipative problems and not encountered in [10]: the projections onto data giving rise to pure TM waves (other than the AH waves) are unbounded. Since these projections are closed, this means mathematically that they cannot reduce S(t) on all of H. Physically, this means that there is some sort of coupling between the TM spatial waves and surface waves. From the structure of the projections we see that this can only occur at high frequencies.

In Section 10 we present the representation for S(t)

S(t)f = I Io/+ exp(- tel 1 i f )R, ,[ + ll&,(t)f + [Is+(t) f + [is_(t)f +

+ ~ [njo.(Of+II~o,.(t)f]+II~.(t)f+IIx,,,(t)f, r > o , i=±1

(0.9)


which for t = 0+ is the Parseval identity for A. The first term on the right is the static part of the solution. The second term is the quasistatic part of the solution corresponding to the eigenvalue -ieTlcr of A. The term l-Ia,(t) is an AH wave consisting of a superposition of AH modes, while II~(t) are surface waves consisting of superpositions of surface modes. Ilioe(t)f and Iljo,.(t)f are, respectively, TE (transverse electric) and TM (transverse magnetic) waves consisting of superpositions of TE and TM plane-wave modes propagating with speed co of the upper medium in opposite directions for j = +1 and j = - 1 . II~,(t)f and IIlm(t)/ are TE and TM waves consisting of superpositions of TE and TM plane-waves modes with the complex frequencies of the lower medium. Here it is not possible to split these waves into modes propagating in opposite directions. For initial data f ~ H (0.9) gives the representation of the solution S(t)f(x)= u(x, t) of problem (1.2). We note that for special initial data the solution (0.9) may consist of a pure wave of any one of the types just mentioned.

The last topic in the paper is a more detailed description of the structure of the surface waves and AH wave. They are given by elementary differential operators applied to solutions of a simple problem for the scalar wave equation in R 3 [13]. We mention that problem (1.2) for TM waves in two spatial variables reduces precisely to this problem for the scalar wave equation in R 2. We also show how data giving rise to pure surface waves or AH waves can be constructed.

1. Mathematical Formulation of the Problem

Let 13 be the 3 x 3 identity matrix, let 03 be the 3 x 3 zero matrix, and let B and E be the diagonal matrices B = diag(-itrI3,03), E(x)= diag(e(x)I3,/x(x)I3) = Xo(X3) diag(eoI3,/~I3) + Xl(X3) diag(elI3,/,1/3) = xoEo+ x1E1. Here Xj(X3) j = 0, 1 are the characteristic functions of the half spaces R~ ={x ~ R 3 : ( - 1 ) J x 3 > 0 } ,

and ej,/~j, tr, respectively, are the positive electromagnetic constants charac- terizing the media filling R 3 ; (r is the conductivity, and we assume that c 2 > c 2, cj = (ej/~) -1/2. Let A(D), Dj = - i 0j, j = 1, 2, 3, be the 6 × 6 matrix operator

3 [ 0 3 /rot] , [ 0 D3 - ~ 2 ] A(D) = ~ AJDj = i rot = - D 3 0 1 (1.1)

- i rot 03 ] j = 1 D2 - D1

We seek a function u(x, t) = t(ul(x, t), uE(x, t)) which is square summable on x E R 3 for each t > 0 and satisfies

i O,U(X, t) ---- {Xo(x3)Eo lA(D) + XI (x3 )ETI[A(D) + B]}u(x , t)

= [Xo(x3)Ao + Xl(x3)A1]u(x, t) = A(D) u(x, t), x3 ~ O, t > O,

u(x, 0+) = f(x) in L 2 ( R 3, C 6)

(1.2)


where f is a prescribed initial function; here u I = '(El, E2, E3) = E is the electric field, and u 2 = '(/41,/-/2,/-/3) = H is the magnetic field. This function is delivered by the contraction semigroup S(t)=exp(-iAt) where the maximal dissipative operator A is defined below.

2. The Maximal Dissipative Operators A, A*

In this section we define operators A, A' which are dissipative in the Hilbert space H consisting of functions f, g in L2(R 3, C 6) with the E inner product

(f' g)= I ;f(x)E(x)g(x)dx, E(x)= Xo(x)Eo+ Xt(x)Et,

where E i are the matrices defined above, and here and henceforth Xi = Xi(x) are the characteristic functions of the half spaces R 3. We then establish that A is maximal dissipative (and thus that A ' - - - E - I [ A - B ] is, in fact, the adjoint A* of A), characterize the eigensubspace for A, A* and their orthogonal complements in H, and obtain the Gillberger decomposition of the space H. Finally, we obtain the interface conditions satisfied by functions in the domain of A (these conditions are used in the construction of the resolvent kernel) and establish the differentiability properties in the tangential directions of functions in the domain of A. For the properties of dissipative operators used below, see, e.g., [14, 17]. We remark that the dissipative operator D of [14, 17] will here be D = - i A , D* = iA', so that the conditions used there here become

O>~Re(f, Df)=Im(f, Af), O>~Re(D*f,f)=Im(A'f,f). (2.1)

Throughout the paper the space of compactly supported, smooth functions on R 3 with values in C 6 is denoted by ~ (R 3, C6).

DEFINITION 2.1. The operator Aw in K =/_a(R 3, C 6) is the operator A of (1.1) on D(Aw)= {f~ K: Af~ K}. The operator As in K is the graph closure of A of (1.1) on ~ ( R 3, C6).

It follows immediately (e.g., by the Fourier transform) that A~ = A*. A simple proof of the following proposition can be found in [11].

PROPOSITION 2.2. As = Aw = A = A*.

DEFINITION 2.3. The operators A °, A, A' in H are defined on D(A °) = D(A) = D(A') = D(A) by

A°f = E-1Af (i.e., A of (0.1) with tr = 0)

Af = A°f + E71x, Bf = XoAof + x,A1 f, (2.2)

A'f = A°f - E~Ix1Bf, f in D(A),

The various parts of the following proposition are obtained via more or less straightforward computations (see [11]).


P R O P O S I T I O N 2.4. (1) The operators A, A' in H are maximal dissipative and A' = A*. The operator A ° is self-adjoint in H.

(2) A necessary condition that f be in D(A) = D(A) = D(A') = D(A °) is that Lf(x', 0+) = Lf(x', 0-) in H-1/2(R 2) (U s is the usual Sobolev space) where

Lf = '(f l , f2, f4, f5) (2.3)

(3) The null spaces N(A), N(A*) of A, A* are closed subspaces, N(A) = N(A*) and the set

So = {'(V0, Vch): O ~ D(R 3, C), ch ~ ~ ( R 3, C)}, Vv = (Dive, D2v2, D3v3),

is dense in N(A). S o = {'(V0, V~b): 0, ~b e ~ ( R 3, C)} is dense in N(A°). (4) Let N,,(A), N~(A*) denote the eigensubspaces corresponding to -io'/el for A

and ioqE1 for A*. Then N~(A) = N~(A*) and the set S~ = if(V0, 0): 0 in D(R31, C)} is dense in N~(A).

(5) Necessary and sufficient conditions that f be in H @ N(A) are that

Xo div fl = 0, Xj div f2 = 0, j = 0, 1 and tZofi(x', 0+) = lx~ fi(x', 0 - ) (2.4)

in H-1/2(R2). Further, a function f is in H @ N , ( A ) if and only if X1 div f l = 0 and we have

f e H Q [ N ( A ) ( ~ N~(A)] if and only if xidiv f k = 0 , j = 0 , 1, k = 1, 2, and (2.4) holds. Finally f is in H O N ( A °) if and only if div(Ef) 1= div(Ef) 2= 0, (i.e., X1 div fk = 0, j = 0, 1, k = 1, 2), (2.4) holds, and

~of~(x', 0+) = El f~(x', 0-).

R E M A R K 2.5. (1) Let f be in ~ ( R 3, C6),

f = f2 , t(fl , f2, f3), t(f4, fs, f6);

integrating by parts, we obtain

(f, A f ) - (Af, f ) = -2io" IR Ifl(x)12 dx. 1

Hence, Im(f, Af)~< 0, and similarly Im(A'f, f)~< 0. Thus, by (2.1) A and A' are dissipative. The operators A, A' can have no proper extensions, since this would imply a symmetric extension of A. Hence, A and A' are maximal, and since A' is a restriction of A* it follows that A' = A*.

(2) As for part two of the proposition, note that f in D(A) implies A3D3f = A f - A 1 D l f - A2D2f is in L2(R, H-I(R2)) and, hence, A 3 f = '(f2, - f l , fs, -f4) is in Cb(R, H-Ue(R2)) [15, 19], where Cb(R, H-1/2(R2)) is the space of bounded, continuous functions from R to H - in (R2), the usual Sobolev space. This is, of course, the usual condition found in the literature regarding continuity of the tangential components of the electric and magnetic fields at the interface. Here


the conditions are automatically satisfied by the requirement that f be in the domain of A.

(3) The remaining parts of Proposition 2.4 can be checked by direct verification.

We observe that the set of smooth functions

T = {E-1A(D)O: 0 ~ ~ ( R 3, C6), supp 0 fq {x3 = 0} = ~b} (2.5)

is contained in /_7/= HO[N(A)(~)N~(A)]. There are other functions of special form i n / 4 which we now construct. Let

[. 1~/24~ be in/ i2 (R 2, C), d(¢) = (~1, ~2, i[~l), 0 = (0, 0, 0), (2.6)

and let x~(l~l) be the characteristic function of the ball { ( : 0 < [ ¢ 1 < R}. We define, for a , /3, X, 6 as in (2.6) and ~x,, the Fourier transform in x',

• x, zr((¢, x3, a) = xR(l¢l)xo(x3)~(~) exp(-xsl¢l)'(d(¢), 0),

• x,Z~(~5, x3,/3) = xR(l~:l)xl(x3)/3(#)exp(x3l~g[)'(a(~:), 0),

~x'Z3R(~ :, X3, X) = XR(I~I)X(~C){~IXo(X3) exp(--x3 I~:l)'(0, d(~))-

- ~X~(x3) exp(x3]~:l)'(0, a(~:))},

• ~,Z~(~, x3, 6) = XR(I~I)Xo(X3)E~6(#) exp(-x3l#l)'(d(~), 0).

(2.7)

Then in the sense of convergence in/-,2 there exist the limits

lim Z~(x; 4,) = Zk(X; ok), (2.8)

(k, ~b) = (1, a), (2,/3), (3, X), (4, 6).

We note that the components of Zk(X; 4~) are harmonic functions on R 3 ; e.g., if 0 is in D(R~, C), then for j = 1 , . . . , 6

(AO, (Z1)j(" ; a)) = lim (A0, (Z~)j( . ; a)) = 0.

We note that the Zk(X; qb) are pairwise orthogonal for (k, ~b)= (1, a), (2, 13), (3, X), and Z4(x; 6) is orthogonal to Z3(x; X).

T H E O R E M 2.6. Let cl(T) be the closure in H of the set T of (2.5), and let [Zk(';~b)] denote the set of all functions of the form (2.7), k = 1 . . . . . 4 wherel" 11/24~ is in Lz(R 2, C). Then cl(T), [Zk ( ' ; ~b)], (k, 4,) = (1, 13), (2,/3), (3, X) are mutually orthogonal subspaces of IZI, cl(T), [Zk(" ; ~b)], (k, 4>) = (3, X), (4, 6), are mutually orthogonal subspaces of IYto = H O N(A°), and there are the following


E-orthogonal direct-sum decompositions:

tQ= el(T) (~)[Z,(. ; ot)](~ [7_a(. ; X)] (~)[Z3(. ;X)],

/~0 = cl(T)~[Z3(" ; X)]G[Z4(' , ~)], (2.9)

H = N(A) (~ N~(A) ~)/-I = N(A °) ~)/-/0.

Moreover, if £, }t °, are the parts of A, A °, in I2I, 171o, then

D(A) VI [Zk(" ; cO)] = {0}, (k, 4,) = (1,/3), (2, fl), (3, X), (2.10)

D(£ °) f'l [Zk(" ; 4,)] = {0}, (k, 4,) = (3, X), (4, 6).

Finally, let k = Zx(', or) + 7-,z(', fl). Then k is in N(A°), and there is the following E-orthogonal direct-sum decomposition:

N(A °) = N(A) (~) No.(A) (~) [ k] (2.11)

and we have

/to = H Q N ( A °) = H O { N ( A ) O N , , ( A ) O [ k ] } .

REMARK 2.7. The decomposition (2.9) is just the familiar Gillberger decomposition for A, A ° in H, see, [10, 11]. Given the explicit representations for the functions in T and [Zk(" ;4')], the assertions of the theorem are obtained by direct verification.

THEOREM 2.8. If f is in D(A °) f-) tqo, then f is in Lz(R, Hi(R2)), and there exists a constant c > O, not depending on f, such that

ID,fl + ID2II ~< clA°fl. (2.12)

If f is in D(A) fl/-), then f need not be in Lz(R, Hi(R2)), but if also f is orthogonal to [k], then f is in D(A °) f'l/)o, and so also in Lz(R, Ha(R2)), and (2.12) holds.

REMARK 2.9. The idea of the proof of (2.12) given in [11] is due to Sarason [18]. A consequence of Proposition 2.4, is that the subspace [k] of Theorem 2.6 is contained i n / 4 and in D(A). If now k = k(. ; a) is any function such that l" I 1/20~ is in Lz(R 2, C) but (.)J I" [ 1/2a is not in Lz(R 2, C), j = 1, 2 then DjK is not in H, and k is not in L2(R, Hi(R2)) (here (.)1 is the function (.)~(¢) = ~). However, if f is in D(A) and i n / 4 but orthogonal to [k], then by (2. l 1) f is orthogonal to N(A °) and, hence, f is in La(R, Hi(R2)), and (2.12) holds. The function k of this theorem is harmonic on R~, j = 0, 1, so it certainly has tangential derivatives, but these derivatives need not belong to L2.

3. The Paramutation Relations, Spectrum, and Resolvents

The maximal dissipative operators A and A* of the preceding section generate continuous semigroups of contraction operators S( t )=exp(- iAt) , S*(t)=


exp(iA*t). Our ultimate goal is to represent S(t) in terms of the generalized eigenfunctions of A, A* and, thus, obtain a representation of the solution u(x, t) = S(t)f(x) of problem (1.2). To this end we need representations of the resolvents G(~')= ( A - ~ I ) -1, G*(~')= (A*-~/) -1 of A, A* in terms of resolvent kernels G(x, y; ~), G*(x, y; ~): for f in H

G(~)f(x) = I G(x, y; ~') f(y) dy,

G*(~)f(x) = I O*(x, y, ~)f(y) dy.

The representations of G, G* are based on explicit formulas for the free space fundamental solutions/j, j = 0, 1 for Aj (Ao = EolA , A1 = E l l [ A + B]) which we now describe.

The transpose of a matrix M is denoted by 'M and the conjugate transpose by ZM, while the adjoint of a matrix or operator is denoted by M*. With respect to the Eo inner product in C 6, ~aEob, a, b c C 6, the symbol Ao(r/), ~/~ R3\{0},

0 -p ~21 [o o ] A o ( ~ ) = ~ 0^ , , 1 ^ = o 0 - l (3 .1)

is self-adjoint with distinct eigenvalues Ajo(n) = jcolnl, j = O, +1, c~) = (~ola.o) -1, each of multiplicity two. The corresponding projections Pio(~/), Ao(7/)Pjo(ag)= )ljo(19)Pjo(~), j = 0, +1, ~9 ~ R 3, are mutually orthogonal orthoprojectors with respect to the Eo inner product: (EoPjo)* = EoPjo, 8jkP~o = PjoPko, I, k = 0, ±1 (Sjk is the Kronecker delta). They are needed explicitly and are given in terms of TM and TE eigenvectors for )ljo, j = 0, ± 1,

'too(r1) = NoM( n) ( - ~lP, - ~zP, I ~12, ~o;toj(n)~:,-Eo;t0j(n)~,, 0),

' eo( n ) = No~. ( n ) ( - ~;to~( n) ~2 , /~Ao~( n) ~, , 0,-~:~p, -~2p, l~12), (3.2)

NoM('q) -t = [~'l[r/l(2~o) 1/2, NoE(*])-' --I~'llnl(2~) '/=, j = +1

'eio(rl) = trll-~(~,, ~2, p, O, O, 0), tmio(~l) = I~1- ' (0 , 0, 0, ~ , , ~=, 0), ~ - - 0,

where ejo, try0 are orthogonal TE and TM vectors (i.e., the normal component of the electric field of ejo and the magnetic field of mio are zero).

Pjo(r/) = mjot[rr~o] + ejo'[eio] j = O, +1.

We observe that the TM and TE vectors are not individually defined for I~[ = 0, but they are bounded in a neighborhood of this line. In terms of these the resolution of the identity for Ao('0) is

I = Poo + P~o + P-~o

and (3.3)

[Ao(n )_ ~i]-1 = ~ [Xjo(n ) _ ~i]-1 pio(Ti). j=O,±l


Let K i denote the space of functions f, g in /_~(R 3, C 6) with the E,- inner product (f, g)j = ([, Ejg) where here and henceforth ( , ) is the usual /_~ inner product with the corresponding norm t[[[I 2 = (L [)- Let .9 0' denote the dual of the space S°(R ") of rapidly decreasing functions.

For any [ in Ko the quantity Ao(D)/ is in SO', and the operator Ao with domain D(Ao)={feKo:Ao(D)fcKo} is self-adjoint in Ko with resolvent Io(~)= (Ao- ~I)-1 given by

Io(~)f(x) = I Io(x, y; ~')[(y) dy, Im ~=/= 0,

where Io(x, y; ~') = Io(x- y; ~) is the fundamental solution,

[Ao-~I]Io(x; ~)= 6(x)I,

obtained from (3.3) by Fourier transform in .9°':

Io(x, y; ~) = (27r) -3 I exp[i(x - y)r/][Ao('r/) - ~1] -1 dr/, (3.4)

Io(Of(x) = O*[A0(" ) - ~.i]-t dpf(x).

In the space K, the operator A , = E ~ I [ A ( D ) + B ] of (2.2) on D(A1) = {f¢ KI:Ey'A(D)f~ K,} is closed and dissipative:

Im(f, Alf) l = - - o r e ~ ' f If'l= <0, f = ,(fl, f2). aR f

The symbol

[-iE-;' o'I -E-fl n^] A,('O) = [ tLq"rl^ 03 J'

has characteristic polynomial given by

det[Al( 'r/)- ffI] = ~'(~" + iey'o')[~- Al1(~)]2[~- A_N(~/)] 2,

with eigenvalues

A0a = 0, A=I = -ie~'o', Ajl(~) = -ie~lo'/2+jA(T1),

X(rt)=[c~Jr/12-(r2e~/4] ltz, ImA(~7)~>0, j = ± l .

and with corresponding eigenvectors

'm,(~/) = N/~(7/)(-¢Ip, --~2P, Ir/I 2, ,1[A,1(~/) + ielltr]~2, --,l[/lkjl(T/) -I'- ietlo-]~,, 0),

'ejl(r/) = N/E(r/)(-/z, Ail(~/)6, ;tlAj,(r/)s~l, 0 , - s ~ , p , - 6 P ,

= , , + i 7' + ie7'o-/2)] 1/2,

Ni,E('r/) -a = tx, ]~[[2etAj,(rt)(Ai,(rt) + ielltr/2)] 1/2, j = ±1,

(3.5)

(3.6)

(3.7)


'eo,(n) = Inl-~(o, 0, 0, ~,, 6 , P),

'm~,(n) = 141-1(6, 6 , p, 0, O, 0).

In terms of these TM and TE eigenvectors we obtain the projections

Pl i (n) + P- , 1(~/) = {e,,(*l)t[Elel,(71)] + e-,,(~l)t[E, e-11(7/)]} +

+{ml,(Tt) '[E,m,,(n)]+ m-, ,(n) '[E~m-,l(~)]} (3.8)

-- e('q)'[Ele(n)] + m(rl)'[E~ m(~/)],

where e(n) = el(,/) + e-l(~/), re(n) = m&,'l) + m-d'rl). The expressions

P01(T/) = eol(n)'[eo,(*l)], Po-l( 'q) = mal( ' r / ) ' [mtr l (T/) ]

are, respectively, the projections corresponding to 0 and -iET~o -. Again we have the resolution of the identity

I = Po,(*/) + P,,ff~/) + Plff'q) + P_~(7/)

and (3.9)

[ a l ( , ~ ) _ ~.i]-1 = ~ (~/.1(7/) _ 0-1pj l ( , lT) , V = {0, o-, +1} . / e v

The terms ejx, rn/1 are not individually defined for {c],/I = or/2~1} whereas the terms e(-r/) and m(,/) are defined there; these terms aren't individually defined at [f] = 0, but they are bounded in a neighborhood of this line. In the expression for [M(7/) - sq] -~ in (3.9) the last two terms are read as

[¢~11('1~) -- ~']-lp11(,Ii ) + [,~._11('!7) - ~']-1p_11(,1~ )

= { - (~+ kr/2e)[Ptff~) + P-l , (n)] + A(7/)[P-u(*/) - Pu(7/)]} x (3.10)

X [X_ll( ' /~) -- ~ ] - l [ ) k l l ( n ) -- ~]-1

in a neighborhood of {¢11,71 =

The spectrum o-(A1) of M in K1 is

o-(A1) = {-irrl2el + X:X in R} U {-iA:A in [0, crib1]} (3.1 1)

where the points 0,-itrlE1 are imbedded eigenvalues. For ~" not in ~r(A1) the resolvent of A~ is given by

h(~ ) f ( x )

I I,(x, y; ~)/(y) dy

= ( I )*[Al (n) - ~ ' I ] - ' ¢l)f(X)

= (I)* E [ ~ k l ( n ) -- ~]-IPkl(~)~f(x),

T H E S O M M E R F E L D P R O B L E M 77

I i(x, y; ¢) = I i(x - y; ¢), (3.12)

I1('; ~) = (2~)-3/2 ~*[AI( • ) - U] -1.

The adjoint A* = ETI[A - B] of A1 has symbol A*(~) = AI(*/) with eigenvalues and corresponding projections which are the complex conjugates of (3.6), (3.9), and its spectrum o-(A*)= ~(A1). For ~ not in o-(A*) the resolvent I~(~)= [A* - 0 ] -1 of A* is given by

I ' l (Of (x ) = I I'l(X, y; ~)/(y) dy (3.13)

= (~*[~k l ( • ) - - ~I]-l@f(x)

= ,v* E f a i l ( n ) - ~] ~ P~,(n)a,f(x),

I'l(x, y; ~-) = I '~ (x - y; ~),

I ; ( . , ~) = (27r)-3/2~*[A~- 0 ] -1. (3.14)

Expressions for //(x, y; ~), I~(x, y; 0 , J = 0, 1 on the hyperplane {x3 = 0} for Y3 : 0 provide the basis for constructing the resolvent kernels for A and A*. The Fourier transform in the x' variables of these traces are obtained using the residue theorem on setting x3 = 0 in the expressions (3.4), (3.12) and (3.14). Simple explicit expressions are obtained for these traces using the explicit formulas in (3.2)-(3.4) and (3.6)-(3.12). We will not present the details of this computation but rather simply describe the essential ingredients and present the final results. For I~1~ ~ 0 the matrix [Ao(~, 1-) - ~I] -1 is regular in the upper ~- half plane except for a pole at the zero of det[Ao(~, ~-)- ¢I], i.e., at ~-= r0 = I-o(¢, ¢) given by

• o(~, ~ ) = co' (¢ ~ -cgl¢l=)~/~; Im I"o>/0, (3.15)

in the ¢ plane with branch cuts (-0% -Col¢l), (Col¢l, oo). Similarly, for I¢1~ ÷ o, the matrices [AI(~, ~-) - ~.I]-1, [A1(¢, ~') - (i]-1 are regular in ~- except for a pole in the upper half plane at ~-=rl = ~1(~, ¢) where for c11~:1 = ~r/2el, denoting by X>(¢), X<(¢) the characteristic functions of the sets {~: c1[¢1 > o-/2el}, {¢: c~[~:[ < ~r/2el}, we have

~-~(¢, ~) = c~a(~(~+ io ' /e ) - ct~l¢12) a/z, Im T1 > 0 , (3.16)

rl(~, ¢) = r>(~, ¢) + ~<(~, 0 ,

here r> = X>rl has branch cuts

(--oo - io'/2e1, [c2[scl 2 - o ' 2 / 4 ~ 2 ] a/2 - io'/2e1),

( [ C 2 [ ~ 1 2 - 0 " 2 / 4 , 2 ] 1 / 2 - io'/2e1 ,oo- io'/2el)

and r< = X<rl with branch cut

( - k r / 2 , t - i[c~l ~:1 = - o-=14,~] 1/2, - io-12,1 + i[c~[¢l 2 - o-2/4 e~]).


Applying q~x, to (3.4), we obtain

• x. to(~, x3, y; 0

i(2¢r) -2 exp(-iy'~) I exp[i(x3- ya)~'][Ao(~, ~')- ~I]- ' dz. (3.17)

Computing Ox'Io(~, 0, y; ~'), Y3 > 0, by the residue theorem, we obtain

~x'Io(~, 0, y; ~)= i(2¢r)-'loexp(-iy'~+ i~'oY3)Po(-~'o), (3.18)

where lo = Co2~"ro 1, and Y3 > 0. Similarly from (3.13), we have

(Isx,Ii(~, x3 ; ~')

exp(-iy'~) I exp[i(x3 - y3)~][AI(~, ,i") - ~-]-1 dr, (3.19) (2~r) -2

• ~,I~(~, x3 ; ~)

= (27r)-1 exp(-iY's¢) I exp[i(x3 - y3)0][Al(~, ~ ) - ~]-1 dr,

and at x3 = 0

d~,,,Ii(~, O, y; ~) = i(2~r)-l/1 e x p ( - i y ' ~ - iy3rl)Pl(rl), (3.20)

• ~,I](~, 0, y; g) = -i(2rr) -1 [1 exp( - iy '~+ iy3?l)/51(-zl),

where 11 = Cl2(~+io'/2el)z~ l, and y 3 < 0 . The projections Pj(+ r~) are given in a very simple form in terms of the

eigenvectors in (3.2), (3.7); defining

m.o = m(~, ~, +to)

= N,,o '( :1: EoX~IT0, q: EO1 ~2'1"0, eol[~[ 2, ~'~z, - ~1 ,0) (3.21)o

e±o = e(~, ~, +to)

= N e o t ( - ~ 2 , f f l , 0, :q~:lt~ol~xTo, ::IT~.Lol~2To,

m±~ = m(~, ~, +rl)

= N.. , '(~: ET'~I~,, :F E7~#~1, ~11#12 , (~ + io'/~,)~, - (~ + i~r/~,)#,, O) (3.21),

e~l = e(s ¢, ~', +r~)

= N e l ' ( - ~ 2 , ~ a , 0, q:/-61'~1Tl, q:~11~27",,/~1-'{~12),

No~ = I t51¢(2~) '/=, NoJ = 1~1~'(2~o) '/=,

mT'., = 1¢112~*,(¢ + i~r/2,,)(~" + iol~,)] ~/~, N~ ) = 1¢112,,¢(¢ + io'12~)] ~/2, Re(.)1/2 > 0.

THE SOMMERFELD PROBLEM 7 9

In terms of (3 .21 ) i , j = 0, 1, the projections P/(+T/) are given by

Pj( :I: ~'i) = PJm(zlz'ri) + Pie( -l- 7i), (3.22)

P#,,( ~ ¢j) = m:~/(Eim±j), Pie( + ~) = e.j'(Eie±j)

and these projections satisfy the following properties

pj( + ~)2 = pj( + ri), Aj(s¢, + ~)pj( ~: ~) = srpj( + ~.j) (3.23)

~,,,Pi, = P~,PJ- = P,,,,Pk, = Pj, P,,,,, = O.

These expressions from the basis for constructing the resolvent kernels for A,

A*. With the funct ions / j and I* of (3.17), (3.19), (3.20), respectively, we seek G,

G' in the form

G(x, y; ~) = Go(x, y; 0 + G,(x, y; 0,

Gi(x , y; ~)= Xi(y3){Xi(x3)[Ii(x , y; ~ ) - Rj(x, y; ~)]+ Xk(X3)Tik(X, y; ~)} (3.24)

where j = 0 , 1, k - - I , 0, k # j .

G'(x, y; ~) = G;(x, y; ~)+ Gi(x, y; b

G~(x, y; if) = Xi(ya){Xj(x3)[I)(x, Y; ~) - R)(x, y; ~)] + T;kfx, y; ()} (3.24)'

(tb(x, y; ~)= lo(x, y; ~). Here, for example, Go(x, y; ~') can be interpreted as the field at the point x due

to a point source at y in R3; the field consists of the incident field I0 and the field Ro reflected from the interface {x3 = 0} if x3 > 0 and of the field To1 transmitted through the interface if x3 < 0. The reflected and transmitted terms should satisfy

0 = [Ao(Dx) - ~I]Ro(x, y; ~') = [Ao(Dx) - ~I]Tao(X, y; s r)

- ; I ] R o ( x , y; ~)=[Ao(O. ) - ( t ]T 'w(x , y; ~), x 3 > 0 , = [Ao(Dx) - ' (3.25)

0 = [ M ( D x ) - ( I lR, (x , y; ~') = [A,(D.) - ~I]Tol(x, y; s r)

= [A*(Dx) - - ' ~I]Rl(x, y; ~)= * - ' [A,(D, , ) -~I]To,(X, y; ~), x3<0 .

By (3.23) conditions (3.25) are satisfied by

¢bx,Ro(~, x3, y; ~)

= i(2 zr)-' lo exp( - iy'~ + iy3 ~o + ix3 to) Po(%) Co,

• ~,T,o(~, x3, y; ~)

= i(2 ~-)-1/o exp(- iy '{~- iy3~'l + ix3ro)Po(~'o)E1EolDo,

• ,,,R'o(~, x3, y; ~-)

= - i (27r)- ' l-o e x p ( - i y ' ~ - iy37o - ix3~o)Po(-ro)Co,


qb~,T'to(~:, x3, y; ~)

= - i(2 ~r)-i io exp(- iy '~ + iy3 ?1 - ix3 ?o)/5o(- ~'o) E1Eo i/)o, (3.26)

~ ,R1(~, x3, y; ~)

= i(2¢r)-Xl~ e x p ( - i y ' ~ - iy3~'~ - ix3~'l)Pl(-~h)C~,

~ 'To l (~ , x3, y; ~)

= i(27r)-1/1 e xp ( - i y '~+ iy3~o- ix3'h)Pl(-'ro)EoE~D1,

~I,~,R~(~, x3, y; {)

= -i(27r)-~ l-~ exp ( - iy '~+ iy3-~ + ix3T1)/51('rl) Cl.

~x'T~a(~, x3, y; ~-)

= -i(27r)-1/-~ e x p ( - i y ' ~ - iy37o + ix3~)P~('ra)EoE11Da,

where the matrices Cj, Dj, etc., must be chosen satisfied:

L~,G(~, 0+, y; ~) = L~x,O(~, 0 - , y; ~)

Lqbx,G'(~:, 0 + , y; ~)= L~x,G'(~, 0 - , y; ~-).

so that condition (2.3) is

(3.27)

(3.27')

The functions in G(Of(x), G*(~)f(x) will then satisfy the necessary condition that they be in D(A), D(A*). We observe that Equations (3.27') are the complex conjugates of (3.27), so that, for example ~R0 contains Co, while qbR~ contains Co. The scalar factors other than the exponential factor containing x3 are chosen for computational convenience. It thus remains to determine the 6 × 6 matrices Cj, D s , j = 0, 1, so that (3.27) is satisfied; (3.27') will then also be satisfied.

The fact that G*(~') is the adjoint of G(~') implies that the resolvent kernels are related by

E(x)G(x, y; 0 = '[E(y)t3*(y, x; ()]. (3.28)

In the self-adjoint case or = 0, G ' = G, and this becomes the familiar relation

E(x)G(x, y; ~)= '[E(y)G(y, x; ()].

From (3.17), (3.19)'

Eolo(x, y; ~) = '[Eo[o(y, x; ~)], EIlI(X, y; ~) = ' [E l / i (y , x; ~)]

and, hence, from (3.24), (3.24)'

EiRi(x, Y; 0 = '[EiR}(Y, x; ~)], (3.29)

EiTjk(X, y; ~) = ' , = [EkTki(y,x;~)], j 0,1, k = l , 0 .

or, equivalently,

EjOx,Ri(~, x3, y; ~) = exp[-i~(x' + y')]'[Es~y,R~(~, Y3, x; ~)], (3.30)

Ei~x.Tjk(~, x3, y; ~) = exp[- i~(x '+ ' ' - ' y )] [EkOy, Tkj(~:, Y3, x; {)] j = 0 , 1, k = 1,0.

Cil?IkO : r]Ml?l_kO ~

loDomko = tMmk l ,

ioDoego = tEek l,


These relations imply that

CjPi(-~) = Ps(~) Cj,

l oDoPo( - Zo) = ll P ( - T , ) E o E 7 ~ D , , (3.31)

loPo(zo)EiEo~Do = l~D~e~(rl).

The relations (3.31) are the reflection and transmission paramutation relations [11].

These expressions are satisfied, in particular, if

Po('ro) Co = roMm+o*(Eom-o) + roEe+o ~(Eoe-o),

loPo('ro) E i E o 1 Do = tMm+o '( E , m+l) + tEe+o ' (El e+x) = ll D1PI(~'I) (3.32)

PI(-T1) C1 ~- r l~m-~ ' (El m+l) + r~Ee-1 t(E1 re+l),

I1P1(- ' r1) E o E 1 1 D 1 = t M m - 1 i(Eotrl-o) q- tee-1 i(Eoe-o) = loDoPo(-'ro).

where rSM, rSE, t~, tE are scalar functions. Recalling the forms of Po and Px in (3.22) we seek

Cieko = riEe_ko ,

liD1 m k l : tMmko ,

l lD lek l = tEekO.

It remains to determine the scalar functions rSM , riE, tM, tE (i.e., the reflection and transmission coefficients) so that (2.3) is satisfied.

Breaking Equation (3.27) into TM and TE parts we have

L m - o = roMLrn+o + lol tMLrn-1, Le-o = roELe+o + lo~ tELe-1 (3.33)0

and

Lm+l = rlMLm-1 + l~ltMLm+o, Le+I = r lELe-i + 171tELe+o (3.33)1

in each of the four equations in two unknowns in (3.33)0 we choose two equations and determine r0M, tM and roe, tE. Thus, choosing, for example, the second and third equations for r0M, tM and the first and fourth for roe, re, we must solve

[~'~,-ol= r-~o'~,-ol, t ,,, , , ,,, ~-I[ Eli ~2TI l ~ 2 J rOM[ (,~2 ] ± M' ' ,Mt 'O' 'OM, [(~.+ io'/el)~21'

(3.34)

p, olf2~.oj = roE[_/~o,f2~.o] + tENiE(loNoE)- ' ~Z 1 '

t o , = ;Xm/Am, t,~ = ~o~I/(I~IAr~No,.,N,M)

roe = A e / A e , tE = p.oP, II(I~IAeNoENIE), (3.35)

am = Eo~'~', - E,(~'+ io ' / e , ) zo , ~e = ~ ' , - t~l ~'o,

Am = eo~r~-i + el(~" + io/e~)~'o, Ae = lXor~ + t~ ~'o. (3.36)


It is readily checked that each of the other pairs of equations in (3.33)0 is also satisfied with these r0M, roE, tM, tE- It is also easy to check that the corresponding equations in (3.33)1 are satisfied with rlM=--r0M, rlE =--roe and tE, tM of (3.35) and (3.36). Thus, with the matrices (3.32) and scalar functions (3.35) and (3.36), the condition (2.3) is satisfied.

Similarly it is easily checked that (3.27)' is satisfied with

[Po(- ~o) ~3o] = [ ~OMfit-O '(Eofit+o) + ~oE~-o'(Eoe÷o)],

[ loPo(-'to)E, Eo' E3o] = [ t-Mfit-0 '(E, fit_,) + l-Ee_0 ' (E, i_,)]

= [ l , D , P l ( - 7 1 ) ] , (3.37)

[P,(T1) C,] ~- [r, Mfit+l ' (E, fit_,) + rlEe+l ' (E l f i t - 1 ) ]

[ IIP,(z,)EoET'19, = tMfit+l '(Eofit+o) + t-~F+, '(Eoff+o)]

= [ loDoPo(+ro)].

We remark that the TM and TE terms just constructed are not individually defined at Ill = 0. For all practical purposes, this makes no difference since they are bounded in a neighborhood of this line. The sums of the TM and TE terms are defined at Ill = 0 . This can be seen directly, but we do this in [11] by exhibiting the explicit form of the matrices Q , Dj.

The expressions for the reflection and transmission coefficients (3.35) and (3.36) are meaningless if ~= ~(f) is a root of A~, s = e, m. Now (see (3.15), (3.16)) for any If[ not zero, the function Im ~'1 = 0 only for ~ in {Im ~ = -kr /~l} O {Re ~ = 0, -of/El < Im ~ < 0}, while Im ~'o = 0 only for Im ~ = 0. Since Im ~ >I 0, it follows that for I f l ¢÷ 0 has no roots. Likewise, in the case o-=0, A,, = ¢(eo~', + Ell'O) has no roots for Ifl ÷ 0. The situation with A,, is quite different in the case o- # 0: it has three roots So(f), s±(f), and for each f in R 2 not zero there corresponds an AH mode (Re So(f) = 0), while to s±(f) there correspond surface modes.

T H E O R E M 3.1. In the case go = tXl, o" not zero for each If] not zero the function

A,. = ~o~T1 + E,(~+ icr/eO'ro

has three roots

So(t) = -is°(t) , s+(t) = -g_(t) = Sl(t) + is2(t),

(3.38)

(3.39)

t = Ill in Ro, where s °, Sl in Ro, S2 in R1. The function s°(t) decreases montonic- ally from s°(0+) = or/el to

sO(o0) = 0"Ellc20(C2--[ - C2) -1, S0t(0+) = --C21~.1[0 -,

s°'(~) = o, s°"(o+) = 2c~,~,~-3(c~o- c~);

thus the range of So(t) is contained in (-io'/~1,-io'~Tlc2(c2+c21)-1). Further, So(If[ 2) lies below the lower branch point of rl for all 0 < c,[f[ < o'/2~, . The real part


sl(t) Of the root s+(t) has the representation

sl( t) = a( t)~/ t, a( t) c (Co, 4(C2o+ c2)),

a2( t) = ~rcZo/ El s°( t) - (s ° - o-/ E1)2 / 4 t

where the function a(t) increases monotonically from co = a(0) to ,J(C2o+ c~)= a@), and

2 Co or Clel(co +c~/4), ol'(~) = O. Ol ' (O) = --1 --1 --2 2 2 2

The imaginary part s2( t) of the roots s~( t) is given by

s2(t) = 2-'[s°(t) - O/E1]

which, thus, decreases monotonically from 0 = s2(0+) to

s2(~) = - 2 - ' , - ' c~( c~ + c~) -1.

4. Representation of the Resoivents

We have now constructed candidates for representations of the resolvents G(~'), G*(0 in the form (3.24), (3.24)' for all points ~ e C\o'(A). Thus, for ~" ~ o-(A)

G(~) f(x) = xoG(~) f(x) + X, G(~) f(x),

XoG(~)f(x)

xo(x3) j{xo(y3)tIo(x, y; ¢)-Ro(x, y; 0]+

+ X,(Y3) Tin(x, y; ~r)}f(y) dy, (4.1)

x, G(Of(x)

= X,(x3) I {X,(Y3)[I6x, y; O - Rffx, y; ~')]+

+ Xo(Y3)Tol(x, y; 0 } f ( y ) d y .

Likewise, for ~ ~ (r(A)

G*(~) f (x) = xoG*(~) f(x) + Xl G*(O f(x),

XoG*(~)f(x)

= Xo(X3) I {Xo(Y3)[I'o(X, y; ~) - R'o(X, y; b]+

+ Xl(y3)T'~o(X, y; ~)}f(y)dy, (4.1)*

x~ G*(Of(x)

= x6x3) I {xl(y3)[I'~(x, y; ~) - R;(x, y; ~)]+

+Xo(y3)T'o6X, y; ~)}f(y)dy.

84

Im(=O

DAVID S. GILLIAM AND JOHN R. SCHULENBERGER

X_lO(q) ~00(~) = kOl(q) - 0 klO(q )

Im ~ = - e l l o / 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x_zl(~)

X_ll(n)

%(0

Fig. 1.

Xll(n) kll(n) ..............................................

X a l ( n ) = - t e ~ l o

THEOREM 4.1. For ~ ¢ o'(A), G(~r), G*(~') are representations of the resolvents for A, A*: for f e H the expressions (4.1) and (4.1)* deliver G(¢) f ~ D(A), G*(~) f e D(A*), and ( A - ~I)G(O f = f, (A* - ~I)G*(~) f = f.

COROLLARY 4.2. The set or(A) is the spectrum of A.

REMARK 4.3. Theorem 4.1 gives an alternative proof that the dissipative operators A, A' are maximal. Figure 1 is a sketch of the spectrum of A. The dotted lines represent the analogues of the curves CN(K) of (0.3). In [11], integration of the resolvent around these curves yields the Parseval identity and complete eigenfunction expansion described below.

5. The Static and Quasistatic Terms

The final forms for IIof, II~,f are given in terms of operators II~, II~, II ± and P by

( Xo(X3)Pfl(x)- II+ f'(x)~ IIo,,X,=\f('~ p[2(x)_Fl-f2(x ) ] (5.1)

n, f(x) = ( x-(x3)Pfl(x)- I] rfl) \ o (5.2)

\ r ! L x ) - n T tx)/


where with c , = (go +/~l)(go + ~1) -1 on 3-vector-valued functions g = t(gl, g2, g3) we have

(I)21-1 ~ g ( ~, x3)

= xo(x3)(21~l) -~ exp(-I~lx3)'d(~)[a(~)~o(~:) + d(sC) Ndsq],

~2HS, g(s ¢, x3)

= xa(x3)(2lscl) -1 exp(]selx3)'3(~)[a(#)go(0 + d(~)g,(~)],

qbzIIg(s ¢, x3) (5.4)

= c+(2l~l)-~[xo(x3) exp(-I~tx3)'d(s ¢) + Xffx3)

x exp(l~Clx3)'d(sq][ + a(s¢) go(~) + d(s¢:)~,(~)],

Pg(x) = ( 2 7 r ) - 3 / 2 I exp(ixr/)p(r/)~3g(r/) dr/, p(r/) = ]r/)-2,r/r/,

gJ(~) = I exp((-1)~+ll~[x3)~2f(~' x3)dx3, d(s c) = (~1, &, i1@.

The important properties of II0, I I , follow easily from more easily justified properties of the above operators; namely,

p2 = p, (II+) 2 = H +, H+P = PII + = I] +, (II-) 2 = H - P = 0,

P I I - = H-.

From these we obtain

(no + n, , ) 2 = no + n ~ , n ~ = r L , rig = no .

T H E O R E M 5.1. The operators Ho, II~ are bounded, self-adjoint projections in H onto the subspaces N(A) = N(A*), N,~(A) = N, dA*), respectively, which reduce A, A*, II,,H, H o H are contained in D(A) and D(A*), and AII~f=-iE71o'I I~f , AIIof = 0.

6. The Surface Wave Modes and Corresponding Projections

A lengthy but more or less straightforward computation yields the following final forms for the surface wave terms Hs..

Define for s = s±

a(~, s) = I~1-2 ~ a m ( ~ , ~')ls = - ( E , G + ~ o G + i2-1eol.tlso-'ros) (6.1)

where s = s:~ and A,,(~, s) = eoS¢~ + effs + iello-)~-o~, ~i~ = ~i(~, s). Further, define

o-(~:, x3) = Xo(X3) exp(i~'osx3)eo(¢o~) - X1(X3)¢o~1~

x exp( - ira ~x3) el (~'1~), (6.2)

tr'(~, x3)= Xo(X3)exp(i¢o~x3)eo(-~'os)- Xl(X3)~-o~'] -1

× exp(-irl~x3)el(-~'l~), (6.3)


eo(~'os) = '(--¢x ~'o,,--¢2"ro,, I~l 2, EoS~2,--eoS¢,, 0),

e l ( 'r , , ) -- ' ( ~ 1 r l , , ~2Tls, I~12, ",(S + io'l'~,),~=,--'~,(S + io'l'~)¢,, 0),

[3s = i'r21sl~l-2( E1T3s + •o'r3s + i2-1EOlZl O'Sros) -1

With this notation we have

T H E O R E M 6.2. I f f e L2(R, Hi(R2)), then

n , ( t ) f ( x ) = a,*(O, exp(-is(Ot)o'(~, x3)fs(~))(x" '),

I ttr'(~' y3)E(y3)~Ef(¢, Y3) dya, f,(O

I L f = IIs(0)f, s = s+.

II~(t) is a bounded map from L2(R, HX(R2)) to H, i.e.,

I n , ( 0 f 12 <- constll [ 1 + ( -a2) 1/2] fLI 2,

and it is closable in H on L2(R, Hi(R2)); the closure is again denoted by II,(t).

Further, IIs( t) satisfies

io,IIs(t)f= An~(t)f, f c ~ ( R 3, C6).

if f ~ D(II,±), then

IL~-f~ D(rI,~_), rL~-f~ D(n~-+),

and

(6.4)

n L f - - n,~f, n,+ns_f = n,_n,+f-- 0.

The closed projections IIs± are not defined on all of H ; in particular, the subspace [k] of Theorem 2.6 is not contained in D(II,±). However, the subspaces N(A) = IIoH, N,(A) = I I~H of Proposition 2.4 are in D(II,±), and

0 = H~±IIoH = IIs±II~H,

0 = nons~D(n,~) = n ~ n ~ D ( n ~ ) .

6.1. THE AH MODES AND ASSOCIATED PROJECTIONS

In the case s = So we have the following

T H E O R E M 6.2. For any f e H a n d s = So, the bounded operator Hs(t) is given by (6.1)-(6.4). ns = II,(0) is a bounded projection in H such that for any f ~ H

0 = n ~ n , f = n o n , f = n ~ n , f = n , n ~ f = n ~ n o f

and for any f e D(II~±), 0 = IIsIIs±f.

THE SOMMERFELD PROBLEM

Further, iO,II~(t)f = AII~(t)f, f e ~ ( R 3, C6).

87

7. The Plane-Wave Modes with Speed Co and the Associated Projections

For smooth, rapidly decreasing f ~ H and j = +1 we define the operators

A)o( n) = a~o( tO) Xo(P)['ajo( to) Eotb3Xo f ( n) -

- rjoa(~l)'aio(~)Eo~aXof(~) + tjla(~)'ajl(f(J)EldP3Xlf(~J). (7.1)

A = M , E , a = m , e .

rjo,,(~) -- aio,,(~)/a/o,~(rt), tjo,,('o) = oJ3[l~:12aio,,(~)N/1,,(xJ)No,,(o/)] -1, (7.2)

Ajo,,(n) = ,onOi 3 + ~10J3 + ij~c01l~l-lo~3,

Ajoe('O) = tzonOJ3 + /"~10")3, n = CoC~ l,

x j = (~, x g , ~ = ( ~ , - p ) , o~ = (,,,1, ,o2 , - , , ,3 ) , ,,, = nil,71 ~ s =-

0~ = -2-1/2411 -10'12]{j(4[1 + ~(,o')lnU] + 1) 1/2+

+ i(4[1 + ,~(,,,')1,71 -=3 - 1) 1/=}

a(,o') = ~2~;2c02(1 -10'12) -2, 0 ' - - n - ' , o '.

The functions eio, mjo, Noe, No,., eja, n~l, Nil, . , Nile are given in (4.2) and (4.7). Similarly, we define the maps E*o from Ko to H by

q~zE*og(~, x3)= (2 ~-)-l/2Xo(x3){ IR+ [e,o(w)exp(i0x3)--

eio(OS) r]oe(r/) exp(--ipx3)] 'ejo(tO)Eog(~, p) dp} + (7.3)

+ (2"n')-l12Xl(X3) IR+ ejl(Xj)~l,(n) ×

× exp(ix]3x3)'ejl(to)Eog(~, p) dp,

and for g ~ Ko I-I Ko~(R 3, C 6) define

dP2M*og(s ¢, x3)----(2~r)-l/2Xo(X3){IR.[mjo(~l)exp(ipx3)--

rn]o('~)rjo,.(r/) exp(-ipx3)] *m~o(n)Eog(~, O) dp} + (7.4)

+ (27r)-'/2X1(X3) [_ rnjl(X])tiz~(rl) X +

X exp(ixi3x3)'~l(to)Eog(~, p) dp,

88

THEOREM 7.1. The operators

IIjo,,(t) = M*oexp(-iAjo(.)t)M)o, t>~O, njom=n,om(0), j = ~ l , (7.5)

are defined on LE(R, Hi(R2)) in H, are bounded from LE(R, Hi(R2)) to H,

IHjom(t)fl <- constll[l + (-A)VE]fll, f ~ L2(R, Hi(R2)),

are closable on LE(R, Hi(R2)) in H with closure again denoted by Hjo,,(t), and they satisfy

iO,Iljora(t)f = AFljom(t)f, f c ~ (R 3, C6),

The operators

l-lio,(t) = Eso exp(-tAio(')t)Ejo, t >~ 0, Ilio, = Hjo,(0), j = ±1, (7.6)

are bounded on H and satisfy

iO,H,oe(t)f= AIIsoe(t) f, f ~ ~ (R 3, C6).

REMARK 7.2. The closures of IIj0m(t) on L2(R, H i ( R E ) ) coincide with their closures on ~ (R 2, C 6) or on rapidly decreasing, smooth functions.

THEOREM 7.3. For j, k = +1, l = 0, or, So, s±

0 = IljomHkOe = Hko~Hjo,. = Hkoe[I1 = HzlIko, = 1- I l I I ,om = Hjor,,Hl

where the notation 0 = A B means that for any f c D(B), B f ~ D(A) and A B f = O.

D A V I D S. G I L L I A M A N D J O H N R . S C H U L E N B E R G E R

8. The Plane-Wave Modes o| Frequencies As1 and the Associated Projections

These terms occasion the greatest technical difficulty. Here the modes cannot be split into plane waves traveling in opposite directions as in the ;tso(~/) case. We present only the final results and refer the reader to [1 1] for the complete details.

For smooth, rapidly decreasing f e H we define the operators

A~, f( n) = aj,( n) Xo(P)[' aj,( ~) EldP3X1 f ( ~l) -

-- rila( ~l)'ajl( ~l)El~3X1 f( ~l) + (8.1)

+ tso,(gl)'ajo(~/)Eo~b3xof(q/J),

A = M , E , a = m , e .

=

tiora( ~l) = CoCl tO3[l~12 A( , l)Ai1,( rl)Njlm( n) Nio,n( Vi)] -1,

~o,(19) = jCoCa tOa[I~]2 Asl( ~)A( 71)Ajl,( rl)Nj~,( ~I) Nsoe( TJ)] -1

a;1.(n) = A('0)(elgb~ + ~onto3)+ i]o(el qbi 3 - ~onto3)/2el, (8.2)

A j t e ( ' r / ) = / . t o n t o 3 - b / z l t ~ , n = CoCT 1,


= (~, ~) , ,~ = (~,-p) ,

No,(~') -a= ~ol~:la,(n)4(2~o),

and for cllnl > cr[2~1

No~(~0 -1= Eo1~1~(~)4(2~o),

89

w h e r e B = M , E a n d b = m , e .

Re ~b~ = -j2-m/z{~/[/32 + aon2(1 -/3)[ n[ -2] +/3 - 2 -1 aol r/I-2} 1/2,

Im 6J3 = 2-1/2{4[/3 2 + aon2(1 -/3)1 rt1-23 - (/3 - 2-~ao1~1-2)} 1/2, (8.3)

ao = o-2/~,eT ', /3 = 1 - n=lo,'l : , a ( n ) ; = c~lnl = - ~r214e2;

while for cllr/I < o/2el

Re 6 ; = 0, (8.4)

Im 6~ -- cTalnl-l lx,(n) 2 - co~l~l=l a/z.

R E M A R K 8.1. Although ~b3 has a proper physical interpretation as an angle only in the case cr = 0, if we interpret Re ~b~ as the angle that the ray transmitted in {x3 > 0} makes with the normal to {x3 = 0} for a plane wave incident on {x3 = 0} in the direction ¢0 = r//Inl from {x3<0}, then we observe that 'total internal reflection' occurs for frequencies cllnl < o-/2el. For a plane wave incident in the direction oJ contained in the cone {oJ : IoJ' 13' c (n -a, 1)} (i.e,, 1/31 = - /3) , Re ~b3 - 0 for large I r/I. Thus, 'almost total internal reflection' again occurs for large Inl and 7/in this cone.

On the sphere cllnl = ~r/2El (i.e., A ( n ) = 0 ) the reflection a n d transmission coefficients as well as the normalization factors in ej~(rl), mia(~q) are singular. In [11] it is shown that the sums

A] = A'll + A'11, A = E, M (8.5)

are nevertheless bounded there. For g • ~ ( R 3, C 6) define

B~*g(x) = B*mg + B*~.g

= f (B~I(X, n) + B*-ll(X, r /))Elg(n) dr/ (8.6) Jn 3

= (27r)-3/2Xl(X3) ..I,,+~ {exp(ixn) ×

x [b11(r/) 'bll(n)Elg(r/) + b-lm(r/)'b-ll(n)Ea g ( n ) ] -

- exp(ixCl)[rlab(n)b11(¢l)'b11(n)Eag(n) +

+ r-l lb(r/)b-H(~) 'b-H(r/)Elg(n)]} dr/+

+ (27r)-3/:X°(x3) _IR1 {exp(ix71)t1°b(n)bl°(71)'blm(~l)Elg(n) +

+ exp(ixv(-1))t_lob(n)b_1o(.y(-1))'bll(n)Etg(r/)} aT


Although M*I, E*I, j = +1, are not individually defined because of the sin- gularity at ;t(r/) = 0 expressions such as

M*l[exp(- iA ( n) t ) - 1]M},, M~j, a(r))M}l

are meaningful. Thus we define

Ill .(t) f(x) = exp(-to/2eO{II, . f (x) + A*l[exp(-i;t(~)t) - 1]A;, f(x) + (8.7)

+ A*n[exp(iAO?)t)- 1]A'_nf(x)},

I l x a f ( X ) = * ' A I A l f ( x ) , for f e H if A = E and f e LE(R, HI(R2)) for A = M.

T H E O R E M 8.2. For t >>- 0 the operators IIl"(t), II~" = IIl,,(O), are bounded from /.~(R, H~(R2)) to H,

I11,,.(t)fl ~< constH[1 + (-A)I/2]fI[, f e L2(R, Hi(R2))

and satisfy

iOtIIl,,(t)f = AlI~,.(t)f. f e ~ ( R 3, C6).

111,.(0 is closable on Lz(R, Hi(R2)) with closure again denoted by [I1,.(0. For t>~O the operators Ille(t), I~le =[Ile(0), are bounded operators on H and

satisfy

iOtI],,(t)f=AII~e(t)f, f e ~ ( R 3, C6).

We have the last of the orthogonality relations

T H E O R E M 8.3. For j, k = +1, 1 = O, a, So, s±

0 = Hl~111e = IIl~IIl,. = HigH1 = nlmII1 = 111Hl,n = H1HI~ = Hl~Hjo~

= H~o.H~e = H~,Hio,. = Hio,. = HI, = H1,.Hjo, = H~o.H~,.

= IIx,.Hio,. = Hio,.HI,..

9. The Final Expression for the Parsevai Equality

Let / : / be the largest common domain of the closed operators IIk, k = s~:, jOin, 1 m, j = + 1. Then / : / con ta ins L2(R, Hi(Re)), but there are functions in / : /wh ich are not in L2(R, Hi(R2)), e.g., general f e (~k~v I I kH) , V={0 , or, So, s±, 10e, --10e, le}.

T H E O R E M 9.1. For the dissipative operator A there is the identity

f=I Io f+I I ,~ f+ ~ Us]f+ ~, (Hio,,f+IIioef)+111ef+IIa,, f, feI?-I, j=O,± j=±l

(9.1)

and this is a direct-sum decomposition of ILl. On their domains all the operators appearing here are projections in H.

T H E SOMMERFELD PROBLEM 91

REMARK 9.2. Equation (9.1) does not hold for all f 6 H. In particular, it holds for all f ~ D(A °) which are orthogonal to N(A°), but only for those f ~ D(A) orthogonal to N(A)@ N,~(A) which are also orthogonal to the subspace [k] of Theorem 2.6. We note, however, that for general f ~ H there is the decomposition

f = Ilof+ rI=f+ rhof+ n ,od+ ILloef + n, f+ Mr,

M= Ilk=f+ ~ (njo,.f+nJ). (9.2) j~=t=l

M is now the bounded projection in H onto data giving rise to propagating TM waves even though the individual terms comprising it are unbounded.

10. The Representation of S ( t ) = exp(-iAt)

We now present the representation of the solution to (1.2).

THEOREM 10.2. For f ~ Lz(R, H1(R2)) S(t) f has the representation

S(t) f=IIof+exp(-to ' /eOH~f+ ~ IIsj(t)f+ ~ (IIjo,,(t)f+ j=0,=t= j==t= 1

+ I I j o e ( t ) f ) +llle(t)f +IIlm(t)f, f ~ t?I, (10.1)

and this is the unique solution of the problem (1.2). The decomposition (10.1) reduces S( t) on 121. For general f ~ H, S( t) f has the representation

S(t) f = Hof+ exp(-t~r/e3H,f+ II,o(t) f + lIxoe(t) f +

+ n_,o~(t)f+ II,e(t) f + M(t)f,

M( t) : II,,,,( t) f + Y. (n,o=( t) f + ns,( t) f ). (10.2) j=± l

and this is the unique solution of problem (1.2) for f ~ D(A). The decomposition (10.2) reduces S(t) on all of H.

We shall now briefly discuss the individual components of the solution. The first term Ilof is the static part of the solution. There is no electrostatic field in R 3, i.e., [ I ] o f ( X ) ] 1 ~- 0 for X3 < 0. The second term exp(-tcr/~OII~f is the quasistatic part of the solution; note that r = E1/0- is the so-called relaxation time for the conducting media filling R13. In this case there is no magnetic part, [II~/] 2= 0, and the electric field is nonzero only in R 3 where it decays exponentially in time. The third component is the surface wave and AH wave terms. For s = So, s±, IIs(t)f(x) is a superposition of TM modes, i.e., [Ils(t)f(x)]6 = 0, which decay exponentially away from the interface {x3=0}, since Im to,, r l~>0. From Theorem 3.1 So(~:) = -is°(£), s°(£) ~ (~r/2el, or/el), so that for s = So II~(t)f(x) is a superposition of modes with frequencies So(0 having no real part which simply


decay in time withoug propagating. This is the AH-wave component of the solution. For s(~)= s±(~)= + s l ( 0 + is2(O, I I s ( t ) f (x ) are the surface-wave components of the solution which consists of modes with frequencies s±(~) that propagate in directions +~ with phase velocity s~(0/[~[ while decaying in time at the rate exp(s2(~)t), s2(~)<0. The fourth and fifth components of the solution consist of TM and TE plane-waves. These waves consist of superpositions of TM and TE plane-wave modes of frequencies )qo('0)= ]Col~ll which in R30 propagate in directions +oJ = +,7/I,71 with phase speed Co. Since X i = (~, Xi3) and Im X~ < 0 in R~ these waves decay exponentially away from the interface {x3 = 0} if or > 0. The components (say wi(x, "1, t)) of these modes in R 3 satisfy ib,wj = ]colaqlwi, so that in this sense they are plane-wave modes propagating with phase speed Co although the interpretation of planes of constant phase and phase velocity in a conducting medium is less relevant (see [26]). The last two components of the solution are spatial TM and TE waves, respectively. These waves are superpositions of the modes exp(-iA_H(T1)t)A_11(x , ~) + exp( - iAn(~ l ) t )A l l (X , ~!), A = E, M coupled on the sphere {cl[r/[ = it/el} which, thus, can not be interpreted as propagating in opposite directions. By (8.3) and (8.4) Im 3'~ > 0 so these modes decay exponentially away from the interface {x3 = 0} in R~) as well as in time. Total internal reflection occurs for Re y~ = 0 as discussed earlier.

11. The Structure o| the Suriace Waves and AH Waves

We now show that for s = So, s~ the surface waves and AH waves S(t)I lsf(x) are essentially scalar waves; They consist of elementary (vector) differential operators applied to solutions of a simple problem for the scalar wave equation [13]. We further show how to construct data [ c D(Hs) giving rise to pure surface waves and how to obtain S( t ) [ from solutions of the aforementioned problem for the scalar wave equation.

Using the notation in Section 6 we define

c~ Lo( ~, x3 , t; f ) = eo exp[ iro~X3 - is( ~) t][3( ~) f , x3 > 0 ,

• 2Lx(~, x3, t; f) = - e l exp[ - i lqsX3- is(0t]/3(~))~ x3 < 0, (11.1)

s(A2) = ),I,2,

'co(D) = ~ 0 1 ( - D 1 D 3 , - D 2 D 3 , - - A 2 , Eos(A2)Dz,--~os(Az)D1,0),

'el(D) = E l l ( - D t D 3 , - D 2 D 3 , - A 2 , EllS(A2)+ itr/E1]D2, - - E I [ S ( A 2 ) "~ for/el]D1,0),

THEOREM 11.1. For f ~ D(II~) s = So, s±

S(t)II~f(x) = Xo(x3)eo(D)Lo(x, t; f ) + Xl(x3)e~(D)Ll(x, t; f ) , (1 1.2)

where L(x , t; f) = go(x3)Lo(x, t; f ) + Xx(x3)Lt(x, t; f ) is the solution of the problem


for the scalar wave equation

(o2_ C2oA)6(x, t) --- O, x~ > O, t > O,

(o2,+o-~;~o,-da)6(x,t)=O, x3<0, t > 0 ,

0 , 6 ( x ' , 0 - , t ) + o - e ~ a q b ( x ' , 0 - , t ) = O , 6 ( x ' , O + , t), t > 0 ,

~'ol03t~(X t, 0"1 t', t) ~--- EllO3~(X ', 0--, t), t > O,

6(x' , o+) = 60, o,4~(x', 0+) = 61(x)

with initial data 49o(X) = L(x , 0+; f) , 4~(x) = OiL(x, 0+; f ) (here, Xj(X3) in the expressions for L are not to be differentiated). I f

(1 + 1~:12)t(O e z,2(R =, C),

po(s ¢, x3, t) = Co exp(il"o~X3 - ist)l(O, x3 > O,

Pl(~, x3, t) = - e l exp(-irl~x3 - ist)rosr~2 1(0, x3 < O,

then

93

(11.3)

of coupe , Me

p(x, t) = xo(x3)po(x, t) + xl(x3)p~(x, t)

is the solution of (11.3) with initial data p(x, 0+), Otp(x, 0+). I f (1 + {~13)l(0 c L2(R 2, C) in the case s = s+, (1 + 1~q3/2)l(~) ~ L2(R 2, C) in the case s = So, then for all t >t 0

Ps(x, t) = Xo(x3)eo(D)po(x, t) + Xl(X3)el( D ) p l ( x , t) (11.4)

is in HsL2(R, Hi (R2) ) , s = s+, I I~H, s = So, and

[ S ( t ) P s ( ' , 0+) ] (x ) = Ps(x, t), s = So, s±,

i.e., P~(x, t) is the solution of (1.2) with initial data P~(x, 0+).

References

1. Bano~, A.: Dipole Radiation in the Presence of a Conducting Half-Space, Pergamon Press, Oxford, 1966.

2. Dunford, N. and Schwartz, J.: Linear Operators, Vol. II, Interscience, New York, 1963. 3. Budden, K. G.: The Wave-Guide Mode Theory of Wave Propagation, Prentice-Hall, New Jersey,

1961. 4. Brekhovskikh, L. M.: Waoes in Layered Media, Academic Press, New York, 1980. 5. Burke, J. J. and Kapany, N. S.: Optical Waoeguides, Academic Press, New York, 1972. 6. Dieudonne, J.: A History of Functional Analysis, North-Holland, Amsterdam, 1981. 7. Felsen, L. B. and Murcuvitz, N.: Radiation Scattering of Waves, Prentice-Hall, New Jersey, 1973. 8. Fok, V. and Leontovich, M.: 'Solution of the Problem of Electromagnetic Waves Along the

Earth's Surface by the Method of the Parabolic Equation', ZhETF 16 (1946), 557-573. 9. Gilliam, D. S.: 'Electromagnetic Waves in a Three Dimensional Asymmetric Dielectric Slab

Waveguide', J. Math. Anal. Appl. (to appear). 10. Gilliam, D. S. and Schulenberger, J. R.: 'Electromagnetic Waves in a Three Dimensional Half

Space with a Dissipative Boundary', J. Math. Anal. Appl. 89 (1982), 129-185.


11. Gilliam, D. S. and Schulenberger, J. R.: 'The Propagation of Electromagnetic Waves Through, Along and Over a Three Dimensional Conducting Half Space', Methoden und Verfahren der mathematischen Physik (to appear).

12. Gilliam D. S. and Schulenberger, J. R.: 'On the Structure of Surface Waves', Adv. in Appl. Math. 4 (1983), 212-243.

13. Gilliam, D. S. and Schulenberger, J. R.: 'A Simple Problem for the Scalar Wave Equation Admitting Surface-Wave and AH-Wave Solutions', ANAH Res. Rep. No, 6 (1982).

14. Goldstein, J. A.: Semigroups of Operators and Abstract Cauchy Problems, Lecture Notes, Tulane University, New Orleans, La., 1970.

15. Lions, J. L. and Magenes, E.: l~obl~mes aux Limites non Homog~nes, Vol. 1, Dunod, Paris, 1968. 16. Loeb, J.: 'Sur les ondes de surface en r~gime transitoire', in J. Brown (ed.), Electromagnetic Wave

Theory, Part 1, Pergamon Press, Oxford, 1967. 17. Phillips, R. S.: 'On Dissipative Operators', in A. K. Aziz (ed.), Lecture Notes in Differential

Equations, Vol. 2, Von Nostrand-Reinhold, Princeton, N. J., 1969. 18. Sarason, L.: 'Remarks on an Inequality of Schulenberger and Wilcox', Ann. Mat. Pur. Appl. 92

(1972), 23-29. 19. Schulenberger, J. R.: 'Boundary Waves on Perfect Conductors', J. Math. Anal. Appl. 66 (1978),

514-549. 20. Schulenberger, J. R.: 'Wave Propagation in Layered Media I: Electromagnetic Waves Over a

Dielectric Clad Perfect Conductor in Three Dimensions', ANAH Corp. Res. Rep. No. 2, 1980. R+ , J. Diff. Eqs. 29 (1978), 405--438. 21. Schulenberger, J. R.: 'Elastic Waves in z,

22. Schulenberger, J. R.: 'The Green's Matrix for Steady State Wave Propagation in a Class of Inhomogeneous Anisotropic Media', Arch. Rat. Mech. Anal. 34, (1969), 380-402.

23. Schulenberger, J. R. and Wilcox, C. H.: 'The Limiting Absorption Principle for Steady-State Wave Propagation in Inhomogeneous Anisotropic Media', Arch. Rat. Mech. Anal. 41 (1971), 46-65.

24. Sommerfeld, A.. "Uber die Ausbreitung der Wellen in der drahtlosen Telegraphie', Ann. Phys. 28 (1909), 665-736.

25. Stone, M. H.: Linear Transformations in Hilbert Space, AMS Colloq. Publ., 1932. 26. Stratton, J.: Electromagnetic Theory, McGraw-Hill, New York, 1941. 27. Wait, J. R.: Electromagnetic Waves in Stratified Media, Pergamon Press, Oxford, 1970. 28. Wilcox, C. H.: 'Steady State Wave Propagation in Homogeneous Anisotropic Media', Arch, Rat.

Mech. Anal. 25, (1967), 201-242. 29. Wilcox, C. H.: 'Transient Wave Propagation in Homogeneous Anisotropic Media', Arch. Rat.

Mech. Anal. 3"/(1970), 323-343. 30. Wilcox, C. H.: 'Transient Electromagnetic Wave Propagation in a Dielectric Waveguide',

Symposia Math. Vol. 18, 1976. 31. Weyl, H.: 'Ausbreitung elektromagnetischer Wellen uber einem ebenen Leiter', Ann. Phys. 60

(1919), 481-500. 32. Zenneck, J.: Wireless Telegraphy, McGraw-Hill, New York, 1915.

spectral analysis of a dissipative problem in electrodynamics: the...

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