Separation of variables in perturbed cylinders
A. Aslanyan, E. B. Davies
TL;DR
The paper addresses the Dirichlet Laplacian on perturbed cylinders and the associated resonance problem in non-compact waveguides. By mapping the domain to a rectangle/strip and applying a Fourier separation in the transverse direction, the authors derive a canonical one-dimensional ODE system for longitudinal mode coefficients and solve it numerically via a stable transfer method. The key contributions are a closed-form, separable reduction to a finite-dimensional ODE system and a robust numerical scheme that handles both eigenvalues and resonances, including near-webbed geometries with narrow throats. The results demonstrate convergence of eigenvalues and resonances to a common limit as the geometry approaches a cusp and show the method's efficiency relative to finite-volume discretisations, making it suitable for ill-conditioned waveguide problems in quantum physics and acoustics.
Abstract
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 non-compact domains. We discuss numerical examples related to quantum waveguide problems.