Measure-geometric Laplacians on the real line

Marc Kesseböhmer, Tony Samuel, Hendrik Weyer

Published 2018-02-13Version 1

We generalise the measure-geometric approach to define a first order differential operator $\nabla_{\eta}$ and a second order differential operator $\Delta_{\eta}$ for measures of the form $\eta \mathrel{:=} \nu + \delta$, where $\nu$ is continuous and $\delta$ is finitely supported; thus extending the program developed by Freiberg and Z{\"a}hle. We determine analytic properties of $\nabla_{\eta}$ and $\Delta_{\eta}$ and show that $\Delta_{\eta}$ is a densely defined, unbounded, linear, self-adjoint operator with compact resolvent. Moreover, we give a systematic way to calculate the eigenvalues and eigenfunctions of $\Delta_{\eta}$. For two leading examples, we determine the eigenvalues and the eigenfunctions, as well as the asymptotic growth rates of the eigenvalue counting function.