A Singular Critical Potential for the Schrödinger Operator
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Bibliographic record
Abstract
Abstract Consider a real potential V on R d , d ≥ 2, and the Schrödinger equation: (LS) i ∂ t u + Δ u − Vu = 0, u ↾ t =0 = u 0 ∈ L 2 . In this paper, we investigate the minimal local regularity of V needed to get local in time dispersive estimates (such as local in time Strichartz estimates or local smoothing effect with gain of 1/2 derivative) on solutions of (LS). Prior works show some dispersive properties when V (small at infinity) is in L d /2 or in spaces just a little larger but with a smallness condition on V (or at least on its negative part). In this work, we prove the critical character of these results by constructing a positive potential V which has compact support, bounded outside 0 and of the order (log | x |) 2 /| x | 2 near 0. The lack of dispersiveness comes from the existence of a sequence of quasimodes for the operator P := −Δ + V . The elementary construction of V consists in sticking together concentrated, truncated potential wells near 0. This yields a potential oscillating with infinite speed and amplitude at 0, such that the operator P admits a sequence of quasi-modes of polynomial order whose support concentrates on the pole.
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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 | 0.002 | 0.010 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.005 | 0.003 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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