Abstract. A new method for solving the diffraction problems on bodies with metallic or semitransparent surfaces, as well as on dielectric bodies is proposed. The method is based on a conception of eigen functions of discrete spectrum, introduced into the problem for open bodies by spectral parameter different from the frequency. These functions satisfy the radiation condition at given real frequency termwise. The method permits to express the solution in the form of discrete series only. Numerical results regarding the two-dimensional open resonator with plane mirrors are presented.

Abstract. Bases of the generalized eigen function method for solving diffraction problems are given. The main property of the method is that the not only frequency can be used as spectral parameter in the homogeneous problem giving rise to a function set for expressing the diffraction field in closed an open domains. The dielectric permittivity of the body, surface impedance or transparency, etc. can be such a parameter. The main advantages of such approach are: a) discreteness of the spectrum in exterior problems; b) simplicity of loss describing in bodies with lossess of only one type; c) decreasing the series dimension in the diffraction field expresions, etc.

Abstract. The functionals stationary on the eigen functions of homogeneous problems arising in the generalized eigen functions method are constructed. The main property of these problems is that the spectral parameter appears in their boundary conditions, or in the equation describing only a part of the volume. The boundary conditions containing spectral parameter are make natural for constructed functionals. Some examples of usage the constructed functionals are presented.

Abstract. The generalized eigen oscillation method is applied to a problem about resonators with small coupling. In the first approximation, the problem of excitation of an open resonator with small hole is solved analytically for arbitrary smooth resonator boundary. Main characteristics of resonator (shift of resonant frequency, Q-factor) and propagation factors of leaky waves in waveguide with the same cross section as the resonator contour are calculated. The coupling of resonators of some concrete forms is investigated.

Abstract. Two problems are solved by the generalized eigen oscillation method. Both they are two-dimensional. The first problem concerns the close H-shaped resonator with dielectric insertion in the passage. In the second problem an open resonator with circular semitransparent outer surface and an elliptic dielectric body inside one are considered. The eigen permittivities (a dielectric permittivity values at which the resonance takes place at given frequency) are calculated and analyzed. In the second problem, the resonator losses can be expressedby the imaginary part of eigen values.

Abstract. A new variant of the generalized eigen oscillations method is proposed, in which the spectral parameter is involved into asymptotical conditions at infinity. This approach can be applied for investigation of open resonators or waveguides with semitransparent walls. In the method, the homogeneous eigen value problem is reduced to a real integral equation. Numerical results for some concrete examples are presented.

Abstract. Main properties of the generalized eigen oscillation method (GEOM) are expounded. The main idea of the method is stated, the comparison of the method with eigen frequency method is made, and some problems are enumerated, to which it is worth while to apply GEOM. Two classes of the problems are considered. In the first class problems the spectral parameter is introduced into the equation. The problems about homogeneous dielectric body in the resonator (close or open), or in the free space belong to this class. A variational approach to solving such problem is described and numerical results concerning some concrete examples are presented. More general problems of the class are the problem about nonhomogeneous dielectric body as well as the quantum mechanics problem of scattering on the quasi-stationary level. Numerical results regarding the last problem are shown. In problems of the second class, the spectral parameter is introduced into boundary conditions or into asymptotical conditions in infinity. The problems about bodies with impedance, metallic or semitransparent boundaries are considered. Open resonators with plane or confocal metallic mirrors, waveguides of arbitrary cross section with longitudinal split, close resonators connected by small split, dielectric resonator with great permittivity value, and open resonator with closed semitransparent wall are considered.

Abstract. Homogeneous boundary value problems arising in two variants of the generalized eigen oscillation method are formulated for the case if the domain is a body of revolution (spectral parameter is the impedance or transparency of the boundary). The angular components of electric and magnetic fields are used as potential function in these problems. Functionals stationary on eigen functions of the problems are obtained.