Free and Harmonic Trapped Spin-1 Bose–Einstein Condensates in \({{\mathbb{R}}^{3}}\)
Bibliographic record
Abstract
Abstract. We investigate physical states of spin-1 Bose–Einstein condensate in [Formula: see text] with mean-field interaction constant [Formula: see text] and spin-exchange interaction constant [Formula: see text], two conserved quantities, the number of atoms [Formula: see text], and the total magnetization [Formula: see text] are involved in. First, in the free case, existence and asymptotic behavior of ground states are analyzed according to the relations among [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. Furthermore, we show that the corresponding standing wave is strongly unstable. When the atoms are trapped in a harmonic potential, we prove the existence of ground states and excited states along with some precisely asymptotics. Besides, we get that the set of ground states is stable under the associated Cauchy flow while the excited state corresponds to a strongly unstable standing wave. Our results not only show some characteristics of three-dimensional spin-1 BEC under the effect between the spin-dependent interaction and the external magnetic field, but also support some experimental observations as well as numerical results on spin-1 BEC.
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.013 | 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".