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

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.