Semigroups for one-dimensional Schrödinger operators with multiplicative Gaussian noise
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Bibliographic record
Abstract
Let H:=−12Δ+V be a one-dimensional continuum Schrödinger operator. Consider Hˆ:=H+ξ, where ξ is a translation invariant Gaussian noise. Under some assumptions on ξ, we prove that if V is locally integrable, bounded below, and grows faster than log at infinity, then the semigroup e−tHˆ is trace class and admits a probabilistic representation via a Feynman-Kac formula. Our result applies to operators acting on the whole line R, the half line (0,∞), or a bounded interval (0,b), with a variety of boundary conditions. Our method of proof consists of a comprehensive generalization of techniques recently developed in the random matrix theory literature to tackle this problem in the special case where Hˆ is the stochastic Airy operator.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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