Non-boundedness of the number of nodal domains of a sum of eigenfunctions
Bibliographic record
Abstract
Generalizing Courant's nodal domain theorem, the ``Extended Courant property'' is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. In the first part of the paper, we prove that the Extended Courant property is false for the subequilateral triangle and for regular $N$-gons ($N$ large), with the Neumann boundary condition. More precisely, we prove that there exists a Neumann eigenfunction $u_k$ of the $N$-gon, with index $4 \le k \le 6$, such that the set $\{u_k \not = 1\}$ has $(N+1)$ connected components. In the second part, we prove that there exist metrics $g$ on $\mathbb{T}^2$ (resp. on $\mathbb{S}^2$) which are arbitrarily close to the flat metric (resp. round metric), and an eigenfunction $f$ of the associated Laplace-Beltrami operator such that the set $\{ f \not = 1 \}$ has infinitely many connected components. In particular the Extended Courant property is false for these closed surfaces. These results are strongly motivated by a recent paper by Buhovsky, Logunov and Sodin (arXiv:1811.03835). As for the positive direction, in Appendix B, we prove that the Extended Courant property is true for the isotropic quantum harmonic oscillator in $\mathbb{R}^2$.
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.007 | 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".