Diffraction Problems and Inversion of Infinite Structured Matrices

Many model diffraction problems, generated by the Helmholtz equation, can be reduced to solving the infinite systems of linear algebraic equations SXsF. It proves that often the operators S in these problems satisfy some operator identities of the form ASySBsP P1 U2, where A and B are diagonal matrices and matrices P1and P2have a finite number of columns. By the rigorous regularization and simultaneous application of the operator identity method one can justify and significantly improve the procedure for solving such systems. The case of the diffraction in the planar waveguide with a cross-sectional jump, absolutely soft upper boundary, and absolutely hard lower boundary is thoroughly considered in the paper as an example. In this case S is an infinite matrix, acting as an unbounded operator. The reflection and transmission coefficients are expressed explicitly via the action of Sy1 on two fixed columns, and the ways of obtaining Sy1 on these columns are investigated