A method for computing waveguide scattering matrices in the presence of discrete spectrum

Abstract

A waveguide G lies in the (n+1)-dimensional Euclidean space for positive integer n, and outside a large ball coincides with the union of finitely many non-overlapping semi-cylinders ("cylindrical ends"). The waveguide is described by the operator {L-μ,B} of an elliptic boundary value problem in G, where L is a matrix differential operator, B is a boundary operator, and μis a spectral parameter. The operator {L,B} is self-adjoint with respect to a Green formula. The role of L can be played, e.g., by the Helmholtz operator, by the operators in elasticity theory and hydrodynamics. As approximation for a row of the scattering matrix S(μ), we take the minimizer of a quadratic functional. To construct the functional, we solve an auxiliary boundary value problem in the bounded domain obtained by truncating the cylindrical ends of the waveguide at distance R. As R tends to indinity, the minimizer a(R,μ) tends with exponential rate to the corresponding row of the scattering matrix uniformly on every finite closed interval of the continuous spectrum not containing the thresholds. Such an interval may contain eigenvalues of the waveguide with eigenfunctions exponentially decaying at infinity ("trapped modes"). Eigenvalues of this sort, as a rule, occur in waveguides of complicated geometry. Therefore, in applications, the possibility to avoid worrying about (probably not detected) trapped modes turns out to be an important advantage of the method.