content_image/368278.jpg content_image/368280.jpg content_image/368279.jpg content_image/368281.jpg content_image/368282.jpg
they are not orthogonal among themselves, but orthogonal to the eigenvectors of the adjoint operator 

which is associated with the electric field generated by a surface distribution of electric dipoles (double layer charge distribution). Therefore, if | u$_{k}$ 〉 is a right eigenvector of M corresponding to eigenvalue χ$_{k}$ , M | u$_{k}$ 〉 = χ$_{k}$ | u$_{k}$ 〉 and 〈 v$_{k}$ ′ | is a left eigenvector corresponding to eigenvalue χ$_{k}$ ′ , 〈 v$_{k}$ ′ | M † = χ$_{k}$ ′ 〈 v$_{k}$ ′ | , then 

with the scalar product defined as the integral over the interface Σ, 

<img src='content_image/368278.jpg'>

<img src='content_image/368280.jpg'>

<img src='content_image/368279.jpg'>

The value 1/2 is always the largest eigenvalue of the operator M , regardless the geometry of the object. This is immediately seen if the object is considered to be conductor ( ε$_{1}$ → ∞ ), and then the interior electric field has to be zero. In that case λ = 1, and the charge density that generates a vanishing internal electric field obeys the equation (1 / 2 − M ) µ = 0, and therefore 1/2 is an eigenvalue of M . However, this eigenmode is not dipole-active and does not contribute to the total polarization of the object. The operator (6) is insensitive to a scale transformation, which means that its eigenvalue and eigenvectors depend only on the shape of the object and not on its size, or electrical properties. 

By employing the spectral representation of the resolvent of the operator M 

the solution of equation (7) is obtained for z = 1 / (2 λ ) as 

<img src='content_image/368281.jpg'>

<img src='content_image/368282.jpg'>

The polarizability of the homogeneous particle is obtained by using the distribution (12) to build the solution (4) of the Laplace equation and use it in equation (2). It has been shown that, operationally, the polarizability is simply the dipole moment of the distribution (12) over unit volume [25, 27] 

