The limiting behaviour of solutions of the acoustic transmission problems

The boundary integral equations for the transmission problems for the scalar Helmholtz equation have the property that the dimension of the null spaces changes as the transmission parameter tends to zero in the case where the wave number is an interior eigenvalue. Therefore, the investigation of the continuous dependence of solutions of the transmission problems leads to a certain singular perturbation problem. In order to investigate this problem, we generalize a perturbation theorem for parameter dependent linear operator equations of the second kind in Banach spaces given by Kress 6, 7]. In our study we also introduce a new integral equation formulation for the transmission problem which is better situated for our purposes than the classical approach.