Asymptotic behavior of the eigenenergies of anharmonic oscillators <i>V</i>(<i>x</i>) = <i>x</i><sup>2N</sup> + <i>bx</i><sup>2</sup>
Bibliographic record
Abstract
Analytic semiclassical energy expansions of the anharmonic oscillator V(x) = x 2N + bx 2 are obtained for arbitrary N. These expressions contain the parameters b and N of the potential explicitly. Analytic expressions for energy level spacing are obtained and used to study the behavior of the eigenenergy level spacing for large energies. These expressions show that asymptotic energy level spacing of the potential V(x) = x 2N + bx 2 increases with the coupling strength b for N = 2 and 3, whereas it decreases for N > 3. Validity of the asymptotic expansions for noninteger N is discussed. PACS Nos.: 03.65Ge, 03.65Sq, 02.30Mv
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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) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".