ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A small spectral gap makes the dimer mode sensitive to perturbations

Example

For each $δ > 0$ , let

$H_{δ} = (\begin{array}{cc} - δ & 0 \\ 0 & δ \end{array}), E_{δ} = (\begin{array}{cc} 0 & δ \\ δ & 0 \end{array}), {\hat{H}}_{δ} = H_{δ} + E_{δ} .$

The exact lowest eigenvector is $e_{1}$ . For every $z \in R^{2}$ , the perturbation satisfies $∥ E_{δ} z ∥_{2} = δ ∥ z ∥_{2}$ , and this amplification tends to zero with the gap, but the lowest eigenvector of ${\hat{H}}_{δ}$ remains at angle $\arctan (\sqrt{2} - 1) > 0$ from $e_{1}$ . Thus an absolute perturbation tolerance alone cannot guarantee a mode error tending to zero as the spectral gap collapses.

Facts & Assumptions

Given: The matrices above and the mode-separation context in A separated dimer mode gives a local inexact saddle error bound.

[F1]

A separated dimer mode gives a local inexact saddle error bound: the mode-angle bound depends on the error-to-gap ratio, not on absolute error alone.

Proof

Given: $δ > 0$ and the two symmetric matrices in the Example.

1.1givenalgebra

The eigenvalues of $H_{δ}$ are $- δ$ and $δ$ , with exact lowest eigenvector $e_{1}$ and spectral gap $2 δ$ . Also $E_{δ}^{T} E_{δ} = δ^{2} I$ , so for every $z$ we have $∥ E_{δ} z ∥_{2}^{2} = z^{T} E_{δ}^{T} E_{δ} z = δ^{2} ∥ z ∥_{2}^{2}$ and hence $∥ E_{δ} z ∥_{2} = δ ∥ z ∥_{2}$ .

1.2givenalgebra

The eigenvalues of ${\hat{H}}_{δ}$ are $\pm \sqrt{2} δ$ . For the negative eigenvalue, the first row equation gives $(- 1 + \sqrt{2}) x + y = 0$ , so a lowest unit eigenvector is proportional to $(1, 1 - \sqrt{2})$ . Its angle with $e_{1}$ is therefore $\arctan (∣ 1 - \sqrt{2} ∣) = \arctan (\sqrt{2} - 1)$ , a positive constant independent of $δ$ .

2.1F1step 1.1step 1.2∎

As $δ ↓ 0$ , the perturbation norm tends to zero while this angle stays fixed at $\arctan (\sqrt{2} - 1) > 0$ . The spectral gap also tends to zero, so this does not contradict the separated-mode bound [F1]; it shows why that bound must include a gap scale.

Depends on

A separated dimer mode gives a local inexact saddle error bound

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Graeme Henkelman and Hannes Jónsson, A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives (standard reference, not scraped)