On the inverse spectral problem for the quasi-periodic Schrödinger equation
Bibliographic record
Abstract
We study the quasi-periodic Schrödinger equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"> <mml:mrow> <mml:mo>-</mml:mo> <mml:msup> <mml:mi>ψ</mml:mi> <mml:mrow> <mml:mo>'</mml:mo> <mml:mo>'</mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>ψ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>E</mml:mi> <mml:mi>ψ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>𝐑</mml:mi> </mml:mrow> </mml:math> in the regime of “small” <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> . Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> </mml:mrow> <mml:mo>'</mml:mo> </mml:msubsup> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo>'</mml:mo> <mml:mo>'</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>𝐙</mml:mi> <mml:mi>ν</mml:mi> </mml:msup> </mml:mrow> </mml:math> , be the standard labeled gaps in the spectrum. Our main result says that if ∈ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo>'</mml:mo> <mml:mo>'</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>-</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mi>m</mml:mi> <mml:mo>'</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>ε</mml:mi> <mml:mo form="prefix">exp</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> </mml:mrow> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>m</mml:mi> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>𝐙</mml:mi> <mml:mi>ν</mml:mi> </mml:msup> </mml:mrow> </mml:math> , with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> being small enough, depending on κ 0 >0 and the frequency vector involved, then the Fourier coefficients of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> obey <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>c</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>m</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>|</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:msup> <mml:mi>ε</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo form="prefix">exp</mml:mo> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> </mml:mrow> <mml:mfrac> <mml:msub> <mml:mi>κ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>m</mml:mi> <mml:mo>|</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>𝐙</mml:mi> <mml:mi>ν</mml:mi> </mml:msup> </mml:mrow> </mml:math> . On the other hand we prove that if | c ( m )|≤ ε exp(− κ 0 | m |) with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> being small enough, depending on κ 0 >0 and the frequency vector involved, then <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mi>m</mml:mi> <mml:mrow> <mml:mo>'</mml:mo> <mml:mo>'</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>-</mml:mo> <mml:msubsup> <mml:mi>E</mml:mi> <mml:mi>m</mml:mi> <mml:mo>'</mml:mo> </mml:msubsup> <mml:mrow> <mml:mo>≤</mml:mo> <mml:mn>2</mml:mn>
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
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 | 0.004 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
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; both teacher heads agree on what is shown here.
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".