Convexity and potential sums for Salpeter-type Hamiltonians
Bibliographic record
Abstract
The semirelativistic Hamiltonian H=βm2+p2+V(r), where V(r) is a central potential in R3, is concave in p2 and convex in p≡p2. This fact enables us to obtain complementary energy bounds for the discrete spectrum of H. By extending the notion of “kinetic potential” we are able to find general energy bounds on the ground-state energy E corresponding to potentials with the form V=∑iaif(i)(r). In the case of sums of powers and the log potential, where V(r)=∑q≠0a(q)sgn(q)r q+a(0)ln(r), the bounds can all be expressed in the semiclassical form E≈minr{βm2+(1/r2)+∑q≠0a(q)sgn(q)(rP(q))q+a(0)ln(rP(0))}. “Upper” and “lower” P numbers are provided for q=−1,1,2, and for the log potential q=0. Some specific examples are discussed, to show the quality of the bounds.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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)
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".