Clarification of the relationship between bound and scattering states in quantum mechanics: Application to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mmultiscripts><mml:mi mathvariant="normal">C</mml:mi><mml:mprescripts/><mml:none/><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:mmultiscripts><mml:mo>+</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:math>
Bibliographic record
Abstract
Using phase-equivalent supersymmetric partner potentials, a general result from the inverse problem in quantum scattering theory is illustrated, i.e., bound-state properties cannot be extracted from the phase shifts of a single partial wave, as a matter of principle. In particular, recent $R$-matrix analyses of the $^{12}\mathrm{C}+\ensuremath{\alpha}$ system, extracting the asymptotic normalization constant of the ${2}^{+}$ subthreshold state ${C}_{12}$, from the $\ensuremath{\ell}=2$ elastic-scattering phase shifts and bound-state energy, are shown to be unreliable. In contrast, this important constant in nuclear astrophysics can be deduced from the simultaneous analysis of the $\ensuremath{\ell}=0,2,4,6$ partial waves in a simplified potential model. A new supersymmetric inversion potential and existing models give ${C}_{12}=144.5\ifmmode\pm\else\textpm\fi{}8.5\ifmmode\times\else\texttimes\fi{}{10}^{3}\phantom{\rule{0.3em}{0ex}}{\text{fm}}^{\ensuremath{-}1∕2}$.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.003 | 0.010 |
| Scholarly communication | 0.004 | 0.013 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.016 | 0.003 |
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 source (direct Gemma or distilled Codex), 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".