On the Solvability of Fredholm Boundary Integral Equations of the First Kind for the Three-Dimensional Transmission Problem on the Spectrum

The paper considers two weakly singular Fredholm boundary integral equations of the firstkind to each of which the three-dimensional Helmholtz transmission problem can be reduced. Theproperties of these equations are studied on the spectra, where they are ill posed. For the firstequation, it is shown that its solution, if it exists on the spectrum, allows finding a solution of thetransmission problem. The second equation in this case always has infinitely many solutions, withonly one of them giving a solution of the transmission problem. The interpolation method forfinding approximate solutions of the integral equations and the transmission problem in questionis discussed.