Asymptotic Distribution of Eigenvalues for a Self-Affine String

TL;DR

This paper investigates how eigenvalues of a self-affine string are distributed asymptotically, establishing and numerically confirming scaling laws for the density of states.

Contribution

It introduces the asymptotic distribution of eigenvalues for a self-affine string and derives scaling laws for the Weyl term.

Findings

Scaling laws for the Weyl term are established.

Numerical confirmation of the asymptotic distribution.

Eigenvalues follow a predictable asymptotic pattern.

Abstract

We consider a string with fixed endpoints where the mass density and/or the elastic coefficient vary in a self-affine way as function of position. It is demonstrated how the eigenvalues in the asymptotic limit are distributed. Scaling laws for the Weyl term of the asymptotic integrated density of states is established and confirmed numerically.