Nodal domains in the square---the Neumann case

Abstract

Å Pleijel has proved that in the case of the Laplacian on the square with Neumann condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues. We identify five Courant sharp eigenvalues for the Neumann Laplacian in the square, and prove that there are no other cases.

arXiv comment27 figures

References in corpus (3)

• A. Stern's analysis of the nodal sets of some families of spherical harmonics revisited
• On the number of nodal domains of the 2D isotropic quantum harmonic oscillator -- an extension of results of A. Stern --
• Eigenfunctions for rectangles with Neumann boundary conditions

Cited by in corpus (4)

• Courant-sharp eigenvalues for the equilateral torus, and for the equilateral triangle
• Courant-sharp eigenvalues of Neumann 2-rep-tiles
• Level sets of certain Neumann eigenfunctions under deformation of Lipschitz domains. Application to the Extended Courant Property
• Courant-sharp eigenvalues of a two-dimensional torus