On the Inverse Spectral Problem for the Quasi-Periodic Schr\\"odinger\n Equation
Bibliographic record
Abstract
We study the quasi-periodic Schr\\"odinger equation $$ -\\psi"(x) + V(x)\n\\psi(x) = E \\psi(x), \\qquad x \\in \\IR $$ in the regime of "small" $V$. Let\n$(E_m',E"_m)$, $m \\in \\zv$, be the standard labeled gaps in the spectrum. Our\nmain result says that if $E"_m - E'_m \\le \\ve \\exp(-\\kappa_0 |m|)$ for all $m\n\\in \\zv$, with $\\ve$ being small enough, depending on $\\kappa_0 > 0$ and the\nfrequency vector involved, then the Fourier coefficients of $V$ obey $|c(m)|\n\\le \\ve^{1/2} \\exp(-\\frac{\\kappa_0}{2} |m|)$ for all $m \\in \\zv$. On the other\nhand we prove that if $|c(m)| \\le \\ve \\exp(-\\kappa_0 |m|)$ with $\\ve$ being\nsmall enough, depending on $\\kappa_0 > 0$ and the frequency vector involved,\nthen $E"_m - E'_m \\le 2 \\ve \\exp(-\\frac{\\kappa_0}{2} |m|)$.\n
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".