This work presents an extensive overview of logarithmic integral operators with kernels depending on one or several complex parameters. Solvability of corresponding boundary value problems and determination of characteristic numbers are analyzed by considering these operators as operator-value functions of appropriate complex (spectral) parameters. Therefore, the method serves as a useful addition to classical approaches. Special attention is given to the analysis of finite-meromorphic operator-valued functions, and explicit formulas for some inverse operators and characteristic numbers are developed, as well as the perturbation technique for the approximate solution of logarithmic integral equations. All essential properties of the generalized single- and double-layer potentials with logarithmic kernels and Green's potentials are considered. Fundamentals of the theory of infinite-matrix summation operators and operator-valued functions are presented, including applications to the solution of logarithmic integral equations.
Many boundary value problems for the two-dimensional Helmholtz equation are discussed and explicit formulas for Green's function of canonical domains with separated logarithmic singularities are presented. This book, presenting all necessary references to the spectral theory, should serve as an efficient instrument for the analysis of logarithmic integral equations and will be of value and interest to researchers in the field of mathematical physics, theoretical electromagnetics, integral equations and spectral theory of operators.
Elements of the theory of integral operators: integral operators with purely logarithmic kernel; integral operators in Holder spaces; logarithmic integral operators and Chebyshev polynomials; integral operators defined on a set of intervals; integral operators with fixed logarithmic singularities; elements of spectral theory; abstract pole pencils; logarithmic integral operators in Sobolev spaces; integral operators with kernels represented by series; methods of small parameter; approximate inversion; approximate semi-inversion. Generalized potentials with logarithmic kernels: generalized potentials; Green's potentials; examples for canonical domains; half-plane; rectangle; circle; exterior of a circle; ring. Summation operators: matrix representation; Galerkin methods and basis of Chebyshev polynomials; summation operators in the spaces of sequences; matrix representation of logarithmic integral operators. Boundary value problems: formulation of the problem; uniqueness and existence theorems; canonical problems - diffraction by strips and slots; diffraction by a slot; diffraction by a strip; diffraction by a screen with a rectangular slotted cavity; scattering by a circular slotted cylinder; eigenoscillations of open and closed slot resonators; closed rectangular slot resonator; open rectangular slot resonator; slotted resonator with circular cross section; the integral and summation equations for the strip problems; summation equations in the problem on eigenfrequencies.