Heat kernel and Riesz transform of Schrödinger operators
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Bibliographic record
Abstract
The goal of this article is two-fold: in the first part, we give a purely analytic proof of the Gaussian estimates for the heat kernel of Schrödinger operators <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Δ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>𝒱</mml:mi> </mml:mrow> </mml:math> whose potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒱</mml:mi> </mml:math> is “small at infinity” in a (weak) integral sense. Our results improve known results that have been proved by probabilistic techniques, and shed light on the hypotheses that are required on the potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒱</mml:mi> </mml:math> . In a second part, we prove sharp boundedness results for the Riesz transform with potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>𝒱</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . A very simple characterization of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -non-parabolicity in terms of lower bounds for the volume growth, which is of independent interest, is also presented.
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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 | 0.000 | 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) | 0.001 | 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