2017/06/30 by James B. Kennedy, Kennedy, J. B.
Computer Science · Mathematics · Physics and Astronomy · #35B05 #35J05 #35P05 #58J50 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1706.10037
openalex publication_date 2017/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce an analogue of Payne's nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of the domain. The assertion here is that any eigenfunction associated with the first nontrivial eigenvalue of the Neumann Laplacian on a domain Ω with rotational symmetry of order two (i.e., x∈Ω iff -x∈Ω) "should normally" be rotationally antisymmetric. We give both positive and negative results which highlight the heuristic similarity of this assertion to the nodal line conjecture, while demonstrating that the extra structure of the problem makes it easier to obtain stronger statements: it is true for all simply connected planar domains, while there is a counterexample domain homeomorphic to a disk with two holes.