In the limit of very thin shells, and using the scaling properties of the operator M , one can show [25, 27] that all four M operators are related to M = M$_{11}$ Equations (16) can then be arranged in a matrix form as By knowing the eigenvectors and the eigenvalues of M the charge densities µ$_{1}$ and µ$_{2}$ can be found by inverting the matrix in (18). For example, µ$_{1}$ is where ˜ λ$_{k}$ = (˜ ε$_{k}$ − ε$_{0}$ ) / (˜ ε$_{k}$ + ε$_{0}$ ) and the equivalent permittivity ˜ ε$_{k}$ is defined for each eigenmode as: where δ = η $^{3}$− 1 ≪ 1. The distribution (20) is similar with the distribution (12) obtained for a homogeneous particle, except that λ has to be replaced for each mode with an equivalent quantity ˜ λ$_{k}$ . Equation (21) can be applied recursively for a multi-shelled structure. The strict separation of electric and geometric properties is weakened in this case, because the shape-dependent eigenvalue χ$_{k}$ appears now in the electric equivalent quantity ˜ λ$_{k}$ . The polarizability of the shelled particle is obtained by using the distribution (20) to build the solution (4) of the Laplace equation and use it in equation (2), to get α = 1 3 1 V$_{1}$ ∑ i,k 〈 x · N$_{i}$ | u$_{k}$ 〉 〈 v$_{k}$ | n · N$_{i}$ 〉 1 / (2 ˜ λ$_{k}$ ) − χ$_{k}$ . (22) ˜ ε$_{k}$ = ε$_{S}$ ( 1 + ε$_{1}$ − ε$_{S}$ ε$_{S}$ + δ (1 / 2 − χ$_{k}$ ) ε$_{1}$ + δ (1 / 2 + χ$_{k}$ ) ϵ$_{S}$ ) , (21) µ$_{e}$ = ∑ k 〈 v$_{k}$ | n · N 〉 (1 / (2 ˜ λ$_{k}$ ) − χ$_{k}$ ) − $^{1}$| u$_{k}$ 〉 , (20)   1 / (2 λ$_{1}$ − M ) (1 / 2 + M ) /η 3 − 1 / 2 + M 1 / (2 λ$_{2}$ ) − M     µ$_{1}$ µ$_{2}$   =  $^{n}$ · N n · $_{N}$  . (18) M$_{12}$ [ µ ] = η − $^{3}$( µ/ 2 + M [ µ ]) M$_{12}$ [ µ ] = − µ/ 2 + M [ µ ] M$_{22}$ [ µ ] = M [ µ ] . (17) 8