Файл: Doicu A., Wriedt T., Eremin Y.A. Light scattering by systems of particles (OS 124, Springer, 2006.pdf

Ei and Hi belong to C0,α(Di,1), and for ki / σ(Di,1), the Maxwell boundaryvalue problem possesses an unique solution.

Theorem 1. Consider Di,1 a bounded domain of class C2 with boundaries S1 and S2. Define the vector potentials

A1(r) = a1(r )g(ki, r, r )dS(r ) ,

S1

A2(r) = a2(r )g(ki, r, r )dS(r )

S2

for a1 L2tan(S1) and a2 L2tan(S2), assume ki / σ(Di,1) and

× × A1 + × × A2 = 0

in Ds and Di,2. Then a1 0 on S1 (a1 vanishes almost everywhere on S1), and a2 0 on S2.

Proof. Defining the electromagnetic fields

j

E = k0εi ( × × A1 + × × A2) ,

j

H = −k0µi × E = × A1 + × A2,

passing to the boundary along a normal direction and using the jump relations for the curl of a vector potential with square integrable density [49], we find that

298 D Completeness of Vector Spherical Wave Functions

(M22a) (r) = n2(r) × × a(r )g(ki, r, r )dS(r ) , r S2 ,

S2

we obtain

1

2 I + M11 a1 + M12a2 = 0,

almost everywhere on S1, and

almost everywhere on S2. Note that M11 and M22 are the singular magnetic operators on the surfaces S1 and S2, respectively, while M12 and M21 are nonsingular operators. In compact operator form, we have

The above integral equation is a Fredholm integral equation of the second kind,

and according to Mikhlin [161] we find that a a0 Ctan(S1,2). Noticing that the operator K map Ctan(S1,2) into C0tan,α(S1,2) and C0tan,α(S1,2) into C0tan,α,d(S1,2),

vector potentials with uniformly H¨older continuous densities, then yields E, H C0,α(Di,1). The jump relations for the double curl of the vector potentials A1 and A2 [49], give n1 × E− = 0 on S1, and n2 × E+ = 0 on S2. Therefore, E and H solve the homogeneous Maxwell boundary-value problem, and since ki / σ(Di,1), it follows that E = H = 0 in Di,1. Finally, from the jump relations (for continuous densities)

n1 × H+ − n1 × H− = a10 = 0,

and

n2 × H+ − n2 × H− = a20 = 0,

we obtain a1 0 on S1, and a2 0 on S2.

The following theorems state the completeness and linear independence of the system of regular and radiating spherical vector wave functions on two enclosing surfaces.

Theorem 2. Let S1 and S2 be two closed surfaces of class C2, with S1 enclosing S2, and let n1 and n2 be the outward normal unit vectors to S1 and S2, respectively. The system of vector functions

n = 1, 2, . . . , m = −n, . . . , n/ki / σ (Di,1)

is complete in L2tan(S1,2).

for n = 1, 2, . . ., and m = −n, . . . , n, yields a1 0 on S1, and a2 0 on S2. As in the proof of Theorem 1, we consider the vector potentials A1 and A2 with densities a1 = n1 × a1, and a2 = n2 × a2, respectively, and define the vector field

j

E = k0εi ( × × A1 + × × A2).

Restricting r to lies inside a sphere enclosed in S2 and using the vector spheri-

cal wave expansion of the dyad gI, we see that the first set of closure relations gives E = 0 in Di,2. Analogously, but restricting r to lies outside a sphere enclosing S1, we deduce that the second set of closure relations yields E = 0 in Ds. Theorem 1 can now be used to conclude.

300 D Completeness of Vector Spherical Wave Functions

Theorem 3. Under the same assumptions as in Theorem 2, the system of vector functions

n = 1, 2, . . . , m = −n, . . . , n/ki / σ (Di,1)

is linearly independent in L2tan(S1,2).

Proof. We need to show that for any Nrank, the relations

give αmn = βmn = γmn = δmn = 0, for n = 1, 2, . . . , Nrank and m = −n, . . . , n. Defining the electromagnetic field

Nrank n

E = αmnM 3mn + βmnN 3mn + γmnM 1mn + δmnN 1mn, n=1 m=−n

we see that n1 × E = 0 on S1 and n2 × E = 0 on S2. The uniqueness of the Maxwell boundary-value problem then gives E = 0 in Di,1, and since E is an analytic function we deduce that E = 0 in Di −{0}. Using the representations for the vector spherical wave functions M 1mn,3 and N 1mn,3 in terms of vector spherical harmonincs mmn and nmn, and the fact that the system of vector spherical harmonics is orthogonal on the unit sphere, we obtain

αmnh(1)n (kir) + γmnjn (kir) = 0,

βmn kirh(1)n (kir) + δmn [kirjn (kir)] = 0,

for r > 0. Taking into account the expressions of the spherical Bessel and Hankel functions for small value of the argument