On the distribution of perturbations of propagated Schrödinger eigenfunctions
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Bibliographic record
Abstract
Let (M,g_0) be a compact Riemmanian manifold of dimension n . Let P_0 (h) \stackrel{\text{def}}= -h^2\Delta_{g}+V be the semiclassical Schrödinger operator for h \in (0,h_0] , and let E be a regular value of its principal symbol. Write \varphi_h for an L^2 -normalized eigenfunction of P_0(h) with eigenvalue E(h) \in [E-o(1),E+ o(1)] . Consider a smooth family of metric perturbations g_u of g_0 with u in the k -ball B^k(\varepsilon) \subset \mathbb R^k of radius \varepsilon>0 . For P_{u}(h) := -h^2 \Delta_{g_u} +V and small |t|>0 , we define the propagated perturbed eigenfunctions \varphi_{h,t}^{(u)}\stackrel{\text{def}}= e^{-\frac{i}{h}t P_u(h) } \varphi_h. They appear in the mathematical description of the Loschmidt echo effect in physics. Motivated by random wave conjectures in quantum chaos, we study the distribution of the real part of the perturbed eigenfunctions regarded as random variables \operatorname{Re} (\varphi^{(\cdot)}_{h,t}(x)): B^{k}(\varepsilon) \to \mathbb R, \quad x\in M. In particular, under an admissibility condition on the metric when (M,g) is chaotic, we compute the h \to 0^+ asymptotics of the variance \text{Var} [\operatorname{Re}(\varphi^{(\cdot)}_{h,t}(x))] and show that the odd moments vanish as h \to 0^+ as long as x is not on the generalized caustic set where V(x)=E .
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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