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 (rectangl e or infinite strip). As a result of this transformation the original eigen value problem is reduced to an equivalent problem for an operator with vari able coefficients. Taking advantage of the simple geometry we separate vari ables by means of the Fourier decomposition method. The ODE system obtained in this way is then solved numerically, yielding the eigenvalues of the op erator. The same approach allows us to find complex resonances arising in s ome noncompact domains. We discuss numerical examples related to quantum wa veguide problems. The aim of these experiments is to compare the method bas ed on the separation of variables with the standard finite-volume procedure . For the most computationally difficult examples related to domains with n arrow throats one can clearly seethe advantages of the proposed method. (C) 2001 Elsevier Science.