Detecting eigenvectors of an operator that are near a specified subspace

TL;DR

This work tackles finding eigenvectors of a self-adjoint operator with compact resolvent that lie near a prescribed subspace. By perturbing the operator along the subspace via $\mathcal{L}(s)=\mathcal{L}+ i s Q$, where $Q$ projects onto $W$, the near-$W$ portion of the spectrum becomes well isolated and amenable to standard spectral methods. The authors prove encoding/decoding theorems that relate eigenpairs of $\mathcal{L}$ and $\mathcal{L}(s)$ with explicit residual bounds, enabling reliable recovery of desired eigenpairs within a user-defined tolerance $\delta^*$. They present a general algorithm template robust to discretization and demonstrate diverse applications, including localized perturbations, approximate symmetries, alternate bases, wave scattering, and graph localization, supported by numerical illustrations. Overall, the framework provides a flexible, efficient route to constrained eigenproblems across physics, engineering, and spectral graph theory, with broad potential impact wherever eigenvectors carrying specific properties are sought.

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.

Figures (3)

• Figure 1: Eigenvector response to a strong, localized perturbation, as discussed in Section \ref{['app:perturbations']}. We show approximate eigenvectors calculated on the domain $\Omega=(-1,1)\times(-1,1)$ for the Schrödinger operator $\mathcal{L} = -\Delta - 18\pi^2\chi_{\Omega/2}$, with Neumann boundary conditions. Here, we apply our algorithm to align eigenvectors with the subspace $W\subset L^2(\Omega)$ of functions that are constant outside $\Omega/2$. We show eigenvectors $\phi_{12}$, $\phi_{13}$, and $\phi_{16}$ above, using two different colorbars; $\phi_{12}$ and $\phi_{16}$ were rejected by our algorithm, while $\phi_{13}$ was accepted. The second colorbar above highlights the difference; even though $\phi_{13}$ is not localized inside the square well, it is approximately constant outside. Notably, all three eigenvectors are rejected by the localization algorithm of Ovall and Reid Ovall2023.
• Figure 2: Eigenvectors with an approximate symmetry, as discussed in Section \ref{['app:symmetry']}. We show approximate eigenvectors of the Laplacian calculated on the domain $\Omega=B_1(0,0)\setminus B_{1/5}(0,1/2)$, with Neumann boundary conditions. Here, we apply our algorithm to align eigenvectors with the subspace $W\subset L^2(\Omega)$ of functions with five-point rotational symmetry. The top three eigenvectors, $\phi_{13}$, $\phi_{32}$, and $\phi_{41}$, are accepted by our algorithm, while the bottom three, $\phi_{16}$, $\phi_{33}$, and $\phi_{52}$, are rejected. Of the bottom three eigenvectors, only $\phi_{52}$ shows a trace of five-point symmetry, and this symmetry breaks down near the deleted region $B_{1/5}(0,1/2)$.
• Figure 3: Localization in a basis of angular momentum states, as discussed in Section \ref{['app:localization']}. Specifically, we show approximate eigenvectors of the Laplacian on the hexagonal annulus shown, with Neumann boundary conditions; we apply our algorithm to align eigenvectors with the subspace $W\subset L^2(\Omega)$ of functions with strictly negative (i.e., clockwise) angular momentum. We show eigenvectors $\psi_{8}$, $\psi_{9}$, and $\psi_{10}$, along with their corresponding angular momenta---the arrowheads on the latter plots indicate direction of probability current. Only the first eigenvector is accepted by our algorithm, reflecting its uniformly negative angular momentum.

Theorems & Definitions (11)

• Example 2.1: Reaction-Diffusion
• Example 2.2: Magnetic Schrödinger
• Proposition 2.3
• proof
• Theorem 2.4
• proof
• Theorem 2.5
• proof
• Theorem 2.6
• proof