Courant's nodal domain theorem for positivity preserving forms
In [Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1923, 81--84 (1923; JFM 49.0342.01)] \textit{R. Courant} proved a result about the nodal domains of the eigenfunctions of a self-adjoint differential operator associated to a quadratic form on a bounded domain of \(\mathbb{R}^d\) with Dirichlet boundary conditions. A nodal domain is a maximal connected set where an eigenfunction does not change its sign. The authors first introduce a notion of nodal domains for positivity preserving forms with compact resolvent, i.e., quadratic forms on an \(L^2\)-space associated to self-adjoint compact resolvents \((L+I)^{-1}\) that map positive functions to positive functions. This notion covers Courant's one and other variations in the literature. Then the authors establish a Courant-type result for these forms.
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