A note on measure-geometric Laplacians (Q331073)

Let \(\mu\) be an atomless Borel probability measure supported on \(K\), where \(\{a, b\}\subset K \subset [a, b]\), for some real numbers \(a < b\), with continuous distribution function \(F_\mu\). The authors study the measure-geometric Laplacians \(\Delta^{\mu}\) with respect to \(\mu\) introduced by \textit{U. Freiberg} and \textit{M. Zähle} [Potential Anal. 16, No. 3, 265--277 (2002; Zbl 1055.28002)]. Set \(\lambda_n := -(\pi n)^2\) for \(n \in {\mathbb N}_0\), \({\mathbb N}_0={\mathbb N}\cup \{0\}\). It is proved that: (i) The eigenvalues of \(\Delta^\mu\) under homogeneous Dirichlet boundary conditions are \(\lambda_n\), for \(n \in {\mathbb N}\), with corresponding eigenfunctions \(f_n^\mu(x)=\sin(\pi n F_\mu(x))\). (ii) The eigenvalues of \(\Delta^\mu\) under homogeneous Neumann boundary conditions are \(\lambda_n\), for \(n \in {\mathbb N}_0\), with corresponding eigenfunctions \(f_n^\mu(x)=\cos(\pi n F_\mu(x))\). Further, it is shown that there exists a measure-geometric Laplacian whose eigenfunctions are the Chebyshev polynomials. The authors illustrate results through specific examples of fractal measures, namely inhomogeneous self-similar Cantor measures and Salem measures.