Spectral Comparison Theorems for the Klein-Gordon equation in d>=1 dimensions
Bibliographic record
Abstract
We first study bound-state solutions of the Klein--Gordon equation φ"(x)=[m^2-〖(E-V(x))〗^2]φ(x), for vector potentials which in one spatial dimension have the form V(x) = vf(x), where f(x)≤0 is the shape of a finite symmetric central potential that is monotone non-decreasing on [0, ∞) and vanishes as x⟶∞, and v>0 is the coupling parameter. \n \nWe characterize the graph of spectral functions of the form v= G(E) which represent solutions of the eigen-problem in the coupling parameter v for a given E: they are concave, and at most uni-modal with a maximum near the lower limit E = -m of the energy E ∈ (-m, m). This formulation of the spectral problem immediately extends to central potentials in d > 1 spatial dimensions. Secondly, for each of the dimension cases, d=1 and d ≥ 2, a comparison theorem is proven, to the effect that if two potential shapes are ordered f_1(r) ≤ f_2(r), then so are the corresponding pairs of spectral functions G_1(E) ≤ G_2(E) for each of the existing eigenvalues. These results remove the restriction to positive energies necessitated by earlier comparison theorems for the Klein--Gordon equation by Hall and Aliyu. Corresponding results are obtained when scalar potentials S(x) are also included. \n \nWe then weaken the condition for the ground states by proving that if ∫_0^x▒〖[f_2 (t)- f_1 (t)]φ(t)dt≥0〗, the corresponding coupling parameters remain ordered, where φ_i= 1, or φ_i is the bound state solution of the Klein--Gordon equation with potential V_i, i = 1, 2. These results are valid for any energy E ∈ (-m, m)., but they are restricted to the ground states. \n \nWe finally present a complete recipe for finding upper and lower spectral bounds for both bounded and unbounded potentials, and we exhibit specific result for the applications for the Woods-Saxon, Gaussian, sech-squared, and Yukawa potentials in dimensions d = 1 and d = 3.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.001 | 0.002 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".