Inverse spectral theory of finite Jacobi matrices
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We solve the following physically motivated problem: to determine all finite Jacobi matrices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J"> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding="application/x-tex">J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and corresponding indices <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i comma j"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">i,j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the Green’s function <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle e Subscript j Baseline comma left-parenthesis z upper I minus upper J right-parenthesis Superscript negative 1 Baseline e Subscript i Baseline mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ⟨ </mml:mo> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mi>I</mml:mi> <mml:mo> − </mml:mo> <mml:mi>J</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false"> ⟩ </mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle e_j,(zI-J)^{-1}e_i\rangle</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> is proportional to an arbitrary prescribed function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis z right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(z)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Our approach is via probability distributions and orthogonal polynomials. We introduce what we call the auxiliary polynomial of a solution in order to factor the map <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper J comma i comma j right-parenthesis long right-arrow from bar left-bracket mathematical left-angle e Subscript j Baseline comma left-parenthesis z upper I minus upper J right-parenthesis Superscript negative 1 Baseline e Subscript i Baseline mathematical right-angle right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>J</mml:mi> <mml:mo>,</mml:mo> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> ⟼ </mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mo fence="false" stretchy="false"> ⟨ </mml:mo> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>z</mml:mi> <mml:mi>I</mml:mi> <mml:mo> − </mml:mo> <mml:mi>J</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false"> ⟩ </mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(J,i,j)\longmapsto [\langle e_j,(zI-J)^{-1}e_i\rangle ]</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> (where square brackets denote the equivalence class consisting of scalar multiples). This enables us to construct the solution set as a fibration over a connected, semi-algebraic coordinate base. The end result is a wealth of explicit constructions for Jacobi matrices. These reveal precise geometric information about the solution set, and provide the basis for new existence theorems.
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.
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 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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