Energy bounds for the spinless Salpeter equation
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Bibliographic record
Abstract
We study the spectrum of the Salpeter Hamiltonian H=βm2+p2+V(r), where V(r) is an attractive central potential in three dimensions. If V(r) is a convex transformation of the Coulomb potential −1/r and a concave transformation of the harmonic-oscillator potential r2, then both upper and lower bounds on the discrete eigenvalues of H can be constructed, which may all be expressed in the form E=minr>0[βm2+P2/r2+V(r)] for suitable values of P here provided. At the critical point r=r̂ the relative growth to the Coulomb potential h(r)=−1/r must be bounded by dV/dh<2β/π.
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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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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 it