Regularly varying functions, generalized contents, and the spectrum of fractal strings

Abstract

We revisit the problem of characterizing the eigenvalue distribution of the Dirichlet-Laplacian on bounded open sets $\Omega\subset\mathbb{R}$ with fractal boundaries. It is well-known from the results of Lapidus and Pomerance \cite{LapPo1} that the asymptotic second term of the eigenvalue counting function can be described in terms of the Minkowski content of the boundary of $\Omega$ provided it exists. He and Lapidus \cite{HeLap2} discussed a remarkable extension of this characterization to sets $\Omega$ with boundaries that are not necessarily Minkowski measurable. They employed so-called generalized Minkowski contents given in terms of gauge functions more general than the usual power functions. The class of valid gauge functions in their theory is characterized by some technical conditions, the geometric meaning and necessity of which is not obvious. Therefore, it is not completely clear how general the approach is and which sets $\Omega$ are covered. Here we revisit these results and put them in the context of regularly varying functions. Using Karamata theory, it is possible to get rid of most of the technical conditions and simplify the proofs given by He and Lapidus, revealing thus even more of the beauty of their results. Further simplifications arise from characterization results for Minkowski contents obtained in \cite{RW13}. We hope our new point of view on these spectral problems will initiate some further investigations of this beautiful theory.