A perturbation problem for transmission eigenvalues in a step-index waveguide

In this paper, we consider the existence of real transmission eigenvalues in a perturbed step-index waveguide. First, the Neumann–Dirichlet operator is defined, transforming the existence problem of transmission eigenvalues into a zero-eigenvalue problem for the operator. It is then proven that the eigenvalue function of this operator is analytic with respect to the parameter k. By employing the Green's function method, the solution form for the boundary value problem of the Helmholtz equation is provided. Subsequently, by exploiting the discontinuity properties of the Green's function, the continuity of the eigenvalue function with respect to the perturbation parameter ϵ is established via the Vitali convergence theorem and the Lebesgue dominated convergence theorem, respectively. Finally, the existence of transmission eigenvalues in the perturbed step-index waveguide is proved.