On the convergence of second order spectra and multiplicity
Provides theoretical justification for using second order spectra to determine eigenvalue multiplicity, addressing a known issue in spectral approximation for numerical analysts.
The paper establishes a theoretical framework showing how second order spectra encode the multiplicity of isolated eigenvalues of self-adjoint operators, supported by numerical experiments on benchmark differential operators.
Let A be a self-adjoint operator acting on a Hilbert space. The notion of second order spectrum of A relative to a given finite-dimensional subspace L has been studied recently in connection with the phenomenon of spectral pollution in the Galerkin method. We establish in this paper a general framework allowing us to determine how the second order spectrum encodes precise information about the multiplicity of the isolated eigenvalues of A. Our theoretical findings are supported by various numerical experiments on the computation of inclusions for eigenvalues of benchmark differential operators via finite element bases.