New infinite families of <i>N</i> th-order superintegrable systems separating in Cartesian coordinates
Bibliographic record
Abstract
Abstract A study is presented of superintegrable quantum systems in two-dimensional Euclidean space E 2 allowing the separation of variables in Cartesian coordinates. In addition to the Hamiltonian H and the second order integral of motion X , responsible for the separation of variables, they allow a third integral that is a polynomial of order N ( N ⩾ 3) in the components p 1 , p 2 of the linear momentum. We focus on doubly exotic potentials, i.e. potentials V ( x , y ) = V 1 ( x ) + V 2 ( y ) where neither V 1 ( x ) nor V 2 ( y ) satisfy any linear ordinary differential equation (ODE). We present two new infinite families of superintegrable systems in E 2 with integrals of order N for which V 1 ( x ) and V 2 ( y ) are given by the solution of a nonlinear ODE that passes the Painlevé test. This was verified for 3 ⩽ N ⩽ 10. We conjecture that this will hold for any doubly exotic potential and for all N , and that moreover the potentials will always actually have the Painlevé property.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.002 | 0.002 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".