A Sinc Method for an Eigenvalue Problem of a Differential Operator with Periodic Coefficients and Its Comparison with Hill's Method

Abstract

We consider a problem of computing spectrum of an ordinary differential operator with periodic coefficients. Due to Floquet's theory, such a problem is reduced to a set of eigenvalue problems for modified operators with a periodic boundary condition. We treat two numerical methods for such problems. A first is Hill's method, which reduces each problem to a matrix eigenvalue problem with the finite Fourier series approximation of eigenfunctions of each operator. This method achieves exponential convergence rate with respect to the size of the matrix. The rate, however, gets worse as the period of the coefficients becomes longer, which is observed in some numerical experiments. Then, in order to realize accurate computation in the cases of the long periods, we propose a second method related to Sinc approximation. Basically, Sinc approximation employs Sinc bases generated by the sinc function sinc(x) = sin(pi x)/(pi x) on R. In this work, a certain variant of the sinc function is adopted to approximate periodic functions. Our method keeps good accuracy in the cases of the long periods, which can be confirmed in some numerical experiments.