Asymptotic behaviour of the spectrum of a waveguide with distant perturbation

We consider the waveguide modelled by n-dimensional tube in which Dirichlet Laplacian is considered perturbed by two distant perturbations. These perturbations are described by two arbitrary abstract operators which are ”localized” in some sense, and the distance between their ”supports” tends to infinity. We study the asymptotic behaviour of the discrete spectrum of such problem. The main results are the convergence theorem and the asymptotics expansions for the eigenvalues. The asymptotic behaviour of the associated eigenfunctions is described as well. We also provide some examples corresponding to the particular choice of the distant perturbations. These examples are potential, second order differential operator, magnetic Schrödinger operator, curved and deformed waveguide, delta interaction, and integral operator.