## 1 Introduction  
The scattering of a quantum electromagnetic (EM) field is related to the wide research area stimulated by potential applications in quantum computing and quantum informatics [1; 2; 3; 4], quantum communications [4; 5], quantum circuits [6], quantum antennas [7; 8; 9; 10; 11; 12; 13], quantum radars and lidars [14; 15; 16], sensing and measurements [17]. Currently, this field is in the initial stages of its development. Probably, it will evolve along a path rather similar to classical electrodynamics and macroscopic antenna theory [18; 19]. On the other hand, from time to time, it is expected to encounter fundamental problems with origins that are completely quantum and that do not have any classical analog (as one of such examples, one may note the manifestation of photon entanglement [1; 2; 3; 4; 5]). One of the first stages in the development of the field consists of the exact analytical solutions of some model problems. This phase has started: the quantization of some exact analytical solutions of Maxwell equations obtained via the separation of variables (spherically layered multimode systems [20] and parabolic mirrors [21]) has been given. As a parallel method of development, one can mention high-frequency physical diffraction theories (for example, quantum diffraction by apertures [22; 23]) and their comparison with the physics of simple quantum-optical devices (beam-splitters, interferometers, etc. [24]). You can expect that this phase will stimulate the establishment of universal numerical techniques, based on the so-called \"discretization\" of Maxwell equations (such as the methods of characteristic modes [25; 26; 27; 28; 29; 30], integral equations [31], finite elements [32], finite differences [33], etc.). In the process of their making, the methods of classical electrodynamics will be an important element of the framework. As a promising step in this direction, one may note the quantization of characteristic modes [34,35]--the preliminary version of integral equations [36]--a numerical approach based on the synthesis of time-domain finite element analysis with the canonical quantization technique [37]. Further progress in the design of quantum devices often necessitates an accurate solution of the problems of quantum field scattering. From a naive point of view, the canonical field quantization looks like a formal replacement of classical fields by the corresponding operators, which act in the Hilbert space of wavefunctions (quantum states) and satisfy appropriate commutation relations [38,39]. However, the problem is not limited to the replacement of classical values with the corresponding field operators. In classical electrodynamics, we note a widely used class of methods (including integral equations), in which the whole system is decomposed into a number of spatial regions. For example, for the scattering from a dielectric body, the integral equation is often formulated with respect to the field inside the body. The outside field is expressed through the inside one via an integral over the body volume. Another example is an antenna, defined as a device which transforms the near field to the far field and vice versa. Such transformation is formally described by special integral relations based on the Green theorems [18]. In the case of quantum field, the situation changes dramatically. The quantum field alongside the field operators (which satisfy Maxwell equations) is characterized by the quantum state (\(\mathbb{F}\)-function). The introduction of partial regions requires a decomposition of the wavefunction of the whole space into those of the partial regions. The observable values are defined as expectation values of the corresponding operators with respect to the quantum state of the system [38,39]. The field operators which are related to the given partial region act on the correspondent partial wavefunction. As a result, the scattering of the quantum EM field will be accompanied by the appearance of entanglement in the new basis.