NODAL SETS AND CONTINUITY OF EIGENFUNCTIONS OF KREIN-FELLER OPERATORS

Let be a compactly supported positive finite Borel measure on R-d. Let 0 < lambda(1) <= lambda(2)<= center dot center dot center dot be eigenvalues of the Krein-Feller operator triangle(mu). We prove that, on a bounded domain, the nodal set of a continuous an-eigenfunction of a Krein-Feller operator divides the domain into at least 2 and at most n + r(n - 1) sub domains, where r(n) is the multiplicity of lambda(n). This work generalizes the nodal set theorem of the classical Laplace operator to Krein-Feller operators on bounded domains. We also prove that on bounded domains on which the classical Green function exists, the eigenfunctions of a Krein-Feller operator are continuous.