On some polynomial potentials in <i>d</i>-dimensions
Bibliographic record
Abstract
The d-dimensional Schrödinger's equation is analyzed with regard to the existence of exact solutions for polynomial potentials. Under certain conditions on the interaction parameters, we show that the polynomial potentials $V_8(r) =\sum _{k=1}^8 \alpha _kr^k, \alpha _8&gt;0$V8(r)=∑k=18αkrk,α8&gt;0 and $V_{10}(r)= \sum _{k=1}^{10} \alpha _kr^k, \alpha _{10}&gt;0$V10(r)=∑k=110αkrk,α10&gt;0 are exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for the existence of these exact solutions are discussed. Finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.
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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) | 0.001 | 0.001 |
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".