Abstract :
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
