Local spectral approximation of unbounded operators: non-asymptotic and unified error quantification for subspace methods

TL;DR

This paper develops a non-asymptotic, unified framework for local spectral approximation of unbounded self-adjoint operators, enabling accurate error quantification and dimension detection despite noise and spectral pollution.

Contribution

It introduces a novel spectral approximation framework using projection-valued measures, applicable to unbounded operators, with rigorous error bounds and noise-robust dimension detection methods.

Findings

Unified non-asymptotic error bounds for spectral approximation

Effective dimension detection in noisy settings

Application to prolate filter diagonalization with accurate spectral predictions

Abstract

We introduce a framework for subspace methods which approximate the spectra of self-adjoint, unbounded operators in a local region. Using the projection-valued measure, we derive integrated spectral inequalities that also apply to unbounded operators. Our framework is non-asymptotic, gap-independent, and enables a unified error quantification of numerical routines subject to multiple error sources. Furthermore, we formalize the class of methods applicable to our framework, and establish a rigorous foundation for dimension detection in the presence of noise as solution to frequent numerical artifacts such as spectral pollution. The practical relevance of this non-asymptotic analysis is substantiated by its recent application to sampled prolate filter diagonalization, where it successfully predicted a sharp accuracy transition linking spectral density to the minimal observation time required to decompose a signal.