Separation of Variables in Deformed Cylinders

We study the Laplace operator subject to Dirichlet boundary conditions in a two-dimensional domain that is one-to-one mapped onto a cylinder (rectangle or infinite strip). As a result of this transformation the original eigenvalue problem is reduced to an equivalent problem for an operator with variable coefficients. Taking advantage of the simple geometry we separate variables by means of the Fourier decomposition method. The ODE system obtained in this way is then solved numerically, yielding the eigenvalues of the operator. The same approach allows us to find complex resonances arising in some noncompact domains. We discuss numerical examples related to quantum waveguide problems. The aim of these experiments is to compare the method based on the separation of variables with the standard finite-volume procedure. For the most computationally difficult examples related to domains with narrow throats one can clearly see the advantages of the proposed method.