Nodal sets and continuity of eigenfunctions of Kreĭ-Feller operators

Let $\mu$ be a compactly supported positive finite Borel measure on ${\mathbb{R}}^d$. Let $0<{\lambda }_1\le {\lambda }_2\le \dots$ be eigenvalues of the Kre$\breve{\mathrm{ı}}$n-Feller operator ${\mathrm{\Delta }}_{\mu }$. We prove that, on a bounded domain, the nodal set of a continuous ${\lambda }_n$-eigenfunction of a Kre$\breve{\mathrm{ı}}$n-Feller operator divides the domain into at least 2 and at most $n+r-1$ subdomains, where $r$ is the multiplicity of ${\lambda }_n$. This work generalizes the nodal set theorem of the classical Laplace operator to Kre$\breve{\mathrm{ı}}$n-Feller operators on bounded domains. We also prove that on bounded domains on which the classical Green function exists, the eigenfunctions of a Kre$\breve{\mathrm{ı}}$n-Feller operator are continuous.