An Approximation Framework for Subspace-based Methods in Spectral Analysis with Accuracy Guarantees

Abstract: A mathematical framework for the approximation of eigenvalues of self-adjoint operators through subspace-based methods is presented. The framework contains spectral inequalities that extend to unbounded operators and account for multiple error sources. We include conceptual remarks, on how such framework addresses contemporary challenges towards a more complete approximation theory for quantum physics. Further analysis considers the computational operation of subspace-based methods and proposes new numerical practices. In particular, analytical guarantees on dimension detection of spectral subspaces in the presence of noise are introduced. The generality of the framework invites application to a broad class of numerical methods, and its utility is demonstrated through recent advances in signal processing.