Convergence of Hill's method for nonselfadjoint operators
Provides a theoretical guarantee for a numerical method used in spectral analysis of periodic operators, now applicable to a broader class including nonselfadjoint cases.
The paper proves convergence of Hill's method for approximating spectra of periodic-coefficient ODE operators, extending to nonselfadjoint operators with symmetric positive definite principal coefficient. The proof uses a generalized Evans function via a 2-modified Fredholm determinant.
By the introduction of a generalized Evans function defined by an appropriate 2-modified Fredholm determinant, we give a simple proof of convergence in location and multiplicity of Hill's method for numerical approximation of spectra of periodic-coefficient ordinary differential operators. Our results apply to operators of nondegenerate type, under the condition that the principal coefficient matrix be symmetric positive definite (automatically satisfied in the scalar case). Notably, this includes a large class of nonselfadjoint operators, which were previously not treated. The case of general coefficients depends on an interesting operator-theoretic question regarding properties of Toeplitz matrices.