Counting eigenvalues of Schrödinger operators using the landscape function
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Bibliographic record
Abstract
We prove an upper and a lower bound on the rank of the spectral projections of the Schrödinger operator -\Delta + V in terms of the volume of the sublevel sets of an effective potential \frac{1}{u} . Here, u is the ‘landscape function’ of G. David, M. Filoche, and S. Mayboroda [Adv. Math. 390 (2021), article no. 107946], namely a solution of (-\Delta + V)u = 1 in \mathbb R^{d} . We prove the result for non-negative potentials satisfying a Kato-type and a doubling condition, in all spatial dimensions, in infinite volume, and show that no coarse-graining is required. Our result yields in particular a necessary and sufficient condition for discreteness of the spectrum. In the case of nonnegative polynomial potentials, we prove that the spectrum is discrete if and only if no directional derivative vanishes identically.
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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) | 0.000 | 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 | 0.001 |
| 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