Density of states engineering: normalized energy density of states band structure using the tridiagonal representation approach
Bibliographic record
Abstract
We expand the quantum mechanical wavefunction in a complete set of orthonormal basis such that the matrix representation of the Hamiltonian is tridiagonal and symmetric. Consequently, the matrix wave equation becomes a symmetric three-term recursion relation for the expansion coefficients of the wavefunction. The solution of this recursion is a set of orthogonal polynomials in the energy whose weight function is the energy density of states of the system. The latter is constructed using Green’s function, which is written as a continued fraction in terms of the Hamiltonian matrix elements. We study the distribution of zeros of the orthogonal polynomials on the real energy line based exclusively on their three-term recursion relations. We show that the zeros are grouped into sets belonging to separated bands on the orthogonality interval. The number of these bands is equal to the periodicity (multiplicity) of the asymptotic values of the recursion coefficients and the location of their boundaries depends only on these asymptotic values. Bound states (if they exist) are located at isolated zeros found in the gaps between the density bands that are stable against variation in the degree of the polynomial for very large degrees. We give examples of systems with a single, double, triple, and quadruple energy density bands.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.006 | 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".