From the electromagnetic point of view various meta-materials, optical crystals, geophysical formations, ice, magnetized plasma, etc., can be described as inhomogeneous anisotropic media with dispersion. The interaction of the electromagnetic field with such media can be studied by different analytical and numerical methods. When a three-dimensional object is of finite extent and is situated in free space, then the method od choice is the volume integral equation (VIE) method, sometimes referred to as the Domain Integral Equation method (mathematical literature) and the Discrete Dipole Approximation (physics). In contrast to one- and two-dimensional cases, where existence of the solution to the scattering problem is a trivial question, in the three-dimensional case solution exists under certain conditions related to the physical properties of the medium in question. Note that the sufficient uniqueness conditions are basically the same for all cases. Previously we have shown that the singular integral operator of the VIE has an essential continuous spectrum, which is given explicitly in terms of the media with dispersion. We shall also prove that in the quasi-static case the discrete eigenvalues are contained within the complex envelope of the continuous spectrum. For anisitropic media with dispersion, especially for magnetized plasma, the question of considareble interest is the stability of such a medium. Within the commonly adopted approach, based on the differential form of the Maxwell's equations and Lorentz or Vlasov's equations of motion, very little can be said about the stability of inhomogeneous objects of finite extent, whereas the VIE formulation is perfectly suited for this taks. Instead of analyzing the stability of the medium itself we propose to analyze the stability of the electromagnetic field in a given medium. Due to self-consistency of the problem the two approaches are in fact identical. Thus, we shall discuss the stability of the field in several practical cases ranging from optical crystals to plasma.@en

We present a simple and unified classification of macroscopic electromagnetic resonances in finite arbitrarily inhomogeneous isotropic dielectric 3D structures situated in free space. By observing the complex-plane dynamics of the spatial spectrum of the volume integral operator as a function of angular frequency and constitutive parameters, we identify and generalize all the usual resonances, including complex plasmons, real laser resonances in media with gain, and real quasistatic resonances in media with negative pertimittivity and gain.@en

Previously we have shown that apart from discrete spectrum (eigenvalues), the threedimensional volume integral operator of electromagnetic scattering has a conitnuous essential spectrum as well. Here we derive the corresponding essential modes. We use the Weyl definition of essential spectrum, where such modes are described in terms of singular sequences. A closed form expression for the required sequence is obtained and its properties are analyzed. We also discuss the physical meaning and the conditions on the excitation of the essential resonances.@en

The spectrum of the volume integral operator of three-dimensional electromagnetic scattering is analyzed. The operator has both continuous essential spectrum, which dominates at lower frequencies, and discrete eigenvalues, which spread out at higher ones. The explicit expression of the operator's symbol provides an exact outline of the essential spectrum for any inhomogeneous anisotropic scatterer with Holder continuous constitutive parameters. Geometrical bounds on the location of discrete eigenvalues are derived for various physical scenarios. Numerical experiments demonstrate good agreement between the predicted spectrum of the operator and the eigenvalues of its discretized version.@en

We show that the mode corresponding to a point of the essential spectrum of the electromagnetic scattering operator is a vector-valued distribution representing the square root fo the three-dimensional Dirac's delta function. An explicit expression for this singular mode in terms of the Weyl sequence is provided and analysed. The essential resonance occurs if the permittivity of an object gets close to zero, which is often the case in plasmas and negative-permittivity metamaterials. Such resonance would lead to a perfect localization (confinement) of the electromagnetic field. Simultaneously, however, a portion of electromagnetic energy is removed from the Hilbert spcae and therefore the whole process may be viewed as absorption.@en