Detecting eigenvectors of an operator that are near a specified subspace

TL;DR

This paper presents an efficient method to identify eigenvectors of a self-adjoint operator that are approximately within a specified subspace, useful for quantum and wave phenomena modeling.

Contribution

It introduces a perturbation-based approach to isolate and find eigenvectors near a subspace, with theoretical bounds and numerical validation.

Findings

The method accurately identifies eigenvectors near a subspace within a specified tolerance.

Perturbation bounds ensure the stability and correctness of the eigenpair identification.

Numerical examples demonstrate the effectiveness of the approach in practical scenarios.

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 self-adjoint operator $\mathcal{L}$ on a Hilbert space $\mathcal{H}$ and a closed subspace $W\subset\mathcal{H}$, can we identify all eigenvectors of $\mathcal{L}$ that lie approximately in $W$? We develop an approach to answer this question efficiently, with a user-defined tolerance and range of eigenvalues, building upon recent work for spatial localization in diffusion operators (Ovall and Reid, 2023). Namely, by perturbing $\mathcal{L}$ 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.