On nodal domains in Euclidean balls (Q2821739)

The paper under review is related to the spectral theory of self-adjoint operators defined by Laplace operator \(-\Delta\) with suitable boundary conditions on the unit ball of \({\mathbb R}^d\). Denoting by \(\lambda_1,\lambda_2,\dots\) the eigenvalues of an operator of this type, one says that one of these eigenvalues \(\lambda_n\) is Courant sharp if there exist a corresponding eigenfunction with exactly \(n\) nodal domains, that is, connected components of the set of zeros of that eigenfunction in the unit ball. The main results are that if \(d=2\) then the Courant sharp eigenvalues of the Neumann Laplacian operator are exactly \(\lambda_1\), \(\lambda_2\), and \(\lambda_4\), while if \(d\geq 3\) then the only Courant-sharp eigenvalues of the Neumann Laplacian operator are \(\lambda_1\) and \(\lambda_2\), and this property is shared by the Dirichlet Laplacian operator.