Potential envelope theory and the local energy theorem
Bibliographic record
Abstract
We consider a one-particle bound quantum mechanical system governed by a Schrödinger operator H=−Δ+v f(r), where f(r) is an attractive central potential and v > 0 is a coupling parameter. If ϕ∈D(H) is a “trial function,” the local energy theorem tells us that the discrete energies of H are bounded by the extreme values of (H ϕ)/ϕ, as a function of r. We suppose that f(r) is a smooth transformation of the form f = g(h), where g is monotone increasing with definite convexity and h(r) is a potential for which the eigenvalues Hn(u) of the operator H=−Δ+u h(r), for appropriate u > 0, are known. It is shown that the eigenfunctions of H provide local-energy trial functions ϕ which necessarily lead to finite eigenvalue approximations that are either lower or upper bounds. This is used to extend the local energy theorem to the case of upper bounds for the excited-state energies when the trial function is chosen to be an eigenfunction of such an operator H. Moreover, we prove that the local-energy approximations obtained are identical to “envelope bounds,” which can be obtained directly from the spectral data Hn(u) without explicit reference to the trial wave functions.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".