A multichannel scheme in smooth scattering theory
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Bibliographic record
Abstract
In this paper we develop the scattering theory for a pair of self-adjoint operators \mbox{ A_{0}=A_{1}\oplus\dots \oplus A_{N} } and A=A_{1}+\dots +A_{N} under the assumption that all pair products A_{j}A_{k} with j\neq k satisfy certain regularity conditions. Roughly speaking, these conditions mean that the products A_{j}A_{k} , j\neq k , can be represented as integral operators with smooth kernels in the spectral representation of the operator A_{0} . We show that the absolutely continuous parts of the operators A_{0} and A are unitarily equivalent. This yields a smooth version of Ismagilov's theorem known earlier in the trace class framework. We also prove that the singular continuous spectrum of the operator A is empty and that its eigenvalues may accumulate only to "thresholds'' of the absolutely continuous spectra of the operators A_{j} . Our approach relies on a system of resolvent equations which can be considered as a generalization of Faddeev's equations for three particle quantum systems.
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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 | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 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 it