Generalized comparison theorems in quantum mechanics
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Bibliographic record
Abstract
This paper is concerned with the discrete spectra of Schrödinger operators H = −Δ + V, where V(r) is an attractive potential in N spatial dimensions. Two principal results are reported for the bottom of the spectrum of H in each angular-momentum subspace ℓ: (i) an optimized lower bound when the potential is a sum of terms V(r) = V(1)(r) + V(2)(r), and the bottoms of the spectra of −Δ + V(1)(r) and −Δ + V(2)(r) in ℓ are known, and (ii) a generalized comparison theorem which predicts spectral ordering when the graphs of the comparison potentials V(1)(r) and V(2)(r) intersect in a controlled way. Pure power-law potentials are studied and an application of the results to the Coulomb-plus-linear potential V(r) = −a/r + br is presented in detail: for this problem an earlier formula for energy bounds is sharpened and generalized to N dimensions.
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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 | 0.000 | 0.000 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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 it