Asymptotic behavior of eigenenergies of nonpolynomial oscillator potentials <i>V</i>(<i>x</i>) = <i>x</i><sup>2<i>N</i></sup> + (<i>λx</i> m1)/(1 + <i>gx</i> m2)
Bibliographic record
Abstract
Analytic semiclassical energy expansions of nonpolynomial oscillator (NPO) potentials V(x) = x2N + (λx[Formula: see text])/(1 + gx[Formula: see text]) are obtained for arbitrary positive integers N, m1, and m2, and the real parameters λ and g using the asymptotic energy expansion (AEE) method. Because the AEE method has been previously developed only for polynomial potentials, the method is extended with new types of recurrence relations. It is then applied to the preceding general NPO to obtain expressions for quantum action variable J in terms of E and the parameters of the potential. These expansions are power series in energy and the coefficients of the series contain parameters λ and g explicitly. To avoid the singularities in the potential we only consider the cases where both λ and g are non-negative at the same time. Using the AEE expressions, it is shown that, for certain classes of NPOs, if potentials have the same N, and the same m1 – m2 or m1 – 2m2 then they have the same asymptotic eigenspectra. It was also shown that for certain cases, both λ and –λ as well as g and –g will produce the same asymptotic energy spectra. Analytic expressions are also derived for asymptotic level spacings of general NPOs in terms of λ and g.
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 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".