Computing Energy Eigenvalues of Anharmonic Oscillators using the Double\n Exponential Sinc collocation Method
Bibliographic record
Abstract
A quantum anharmonic oscillator is defined by the Hamiltonian ${\\cal H}=\n-\\frac{ {\\rm d^{2}}}{{\\rm d}x^{2}} + V(x)$, where the potential is given by\n$V(x) = \\sum_{i=1}^{m} c_{i} x^{2i}$ with $c_{m}>0$. Using the Sinc collocation\nmethod combined with the double exponential transformation, we develop a method\nto efficiently compute highly accurate approximations of energy eigenvalues for\nanharmonic oscillators. Convergence properties of the proposed method are\npresented. Using the principle of minimal sensitivity, we introduce an\nalternate expression for the mesh size for the Sinc collocation method which\nimproves considerably the accuracy in computing eigenvalues for potentials with\nmultiple wells.\n We apply our method to a number of potentials including potentials with\nmultiple wells. The numerical results section clearly illustrates the high\nefficiency and accuracy of the proposed method. All our codes are written using\nthe programming language Julia and are available upon request.\n
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| 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".