Published In

Mathematics of Computation

Document Type

Pre-Print

Publication Date

4-23-2026

Subjects

Eigenvectors

Abstract

In modeling quantum systems or wave phenomena, one is often interested in identifying eigenstates that approximately carry a specified property; scattering states approximately align with incoming and outgoing traveling waves, for instance, and electron states in molecules often approximately align with superpositions of simple atomic orbitals. These examples—and many others—can be formulated as the following eigenproblem: given a selfadjoint operator L on a Hilbert space H and a closed subspace W ⊂ H, can we identify all eigenvectors of L that lie approximately in W? We develop an approach to answer this question efficiently, with a userdefined tolerance and range of eigenvalues, building upon recent work for spatial localization in diffusion operators (see J. S. Ovall and R. Reid [Math. Comp. 92 (2023), pp. 1005–1031]). Namely, by perturbing L appropriately along the subspace W, we collect the eigenvectors near W into a well-isolated region of the spectrum, which can then be explored using any of several existing methods. We prove key bounds on perturbations of both eigenvalues and eigenvectors, showing that our algorithm correctly identifies desired eigenpairs, and we support our results with several numerical examples.

Rights

© Copyright the author(s) 2026

Description

This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published as:  (2026). Detecting eigenvectors of an operator that are near a specified subspace. Mathematics of Computation. Portico.

DOI

10.1090/mcom/4101

Persistent Identifier

https://archives.pdx.edu/ds/psu/44663

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