Nodal sets and continuity of eigenfunctions of Krein-Feller operators

Let µ be a compactly supported positive finite Borel measure on R^d. Let 0 < λ₁ ≤ λ₂ ≤ · · · be eigenvalues of the Kreĭn-Feller operator ∆_µ. We prove that, on a bounded domain, the nodal set of a continuous λ_n-eigenfunction of a Kreĭn-Feller operator divides the domain into at least 2 and at most n + r_n − 1 subdomains, where r_n is the multiplicity of λ_n. This work generalizes the nodal set theorem of the classical Laplace operator to Kreĭn-Feller operators on bounded domains. We also prove that on bounded domains on which the classical Green function exists, the eigenfunctions of a Kreĭn-Feller operator are continuous.