Nodal sets and continuity of eigenfunctions of Kre\u{\i}-Feller operators

TL;DR

This paper extends the classical nodal set theorem to Kreeller operators, showing how eigenfunctions' nodal sets partition domains and establishing their continuity under certain conditions.

Contribution

It generalizes the classical Laplace operator's nodal set theorem to Kreeller operators and proves eigenfunction continuity on domains with classical Green functions.

Findings

Nodal set divides the domain into at least 2 and at most n+r-1 subdomains.

Eigenfunctions are continuous on domains with classical Green functions.

Generalization of classical Laplace eigenfunction properties to Kreeller operators.

Abstract

Let $\mu$ be a compactly supported positive finite Borel measure on $\R^{d}$. Let $0<\lambda_{1}\leq\lambda_{2}\leq\ldots$ be eigenvalues of the Kre$\breve{{\i}}$n-Feller operator $\Delta_{\mu}$. We prove that, on a bounded domain, the nodal set of a continuous $\lambda_{n}$-eigenfunction of a Kre$\breve{{\i}}$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$\breve{{\i}}$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$\breve{{\i}}$n-Feller operator are continuous.