Linear adiabatic dynamics generated by operators with continuous spectrum. I
Bibliographic record
Abstract
We are interested in the asymptotic behavior of the solution to the Cauchy problem for the linear evolution equation iε ∂ t ψ=A(t)ψ, A(t)=A 0 +V(t), ψ(0)=ψ 0 , in the limit ε→0. A case of special interest is when the operator A(t) has continuous spectrum and the initial data ψ 0 is, in particular, an improper eigenfunction of the continuous spectrum of A(0). Under suitable assumptions on A(t), we derive a formal asymptotic solution of the problem whose leading order has an explicit representation. A key ingredient is a reduction of the original Cauchy problem to the study of the semiclassical pseudo-differential operator ℳ=M(t, iε ∂ t ) with compact operator-valued symbol M(t, E)=V 1 (t)(A 0 −EI) −1 V 2 (t), V(t)=V 2 (t)V 1 (t), and an asymptotic analysis of its spectral properties. We illustrate our approach with a detailed presentation of the example of the Schrödinger equation on the axis with the δ-function potential: A(t)=−∂ xx +α(t)δ(x).
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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 | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".