Generalised Krein-Feller operators and gap diffusions via transformations of measure spaces

TL;DR

This paper introduces a method to analyze generalized Krein-Feller operators by transforming measure spaces, enabling the transfer of eigenvalue problems to classical settings and providing new insights into spectral dimensions and gap diffusions.

Contribution

It establishes an isometric isomorphism that relates generalized Krein-Feller operators to classical ones, offering a novel characterization of spectral properties and unifying existing results.

Findings

Eigenvalue problems can be transferred via measure space transformations.

New characterization of the upper spectral dimension.

Connections to generalized gap diffusions are clarified.

Abstract

We consider the generalised Krein-Feller operator $\Delta_{\nu, \mu} $ with respect to compactly supported Borel probability measures $\mu$ and $\nu$ with the natural restrictions that $\mu$ is atomless, the supp$(\nu)\subseteq$supp$(\mu)$ and the atoms of $\nu $ are embedded in the supp$(\mu)$. We show that the solutions of the eigenvalue problem for $\Delta_{\nu, \mu} $ can be transferred to the corresponding problem for the classical Krein-Feller operator $\Delta_{\nu \circ F_{\mu}^{-1}, \Lambda}$ with respect to the Lebesgue measure $\Lambda$ via an isometric isomorphism determined by the distribution function $F_\mu$ of $\mu$. In this way, we obtain a new characterisation of the upper spectral dimension and consolidate many known results on the spectral asymptotics of Krein-Feller operators. We also recover known properties of and connections to generalised gap diffusions associated to these operators.