Asymptotics of eigenvalues of high-order differential operator with discrete self-similar weight

TL;DR

This paper investigates the asymptotic behavior of eigenvalues for high-order differential operators with singular, self-similar weights, revealing exponential growth influenced by operator order and self-similarity parameters.

Contribution

It provides the first asymptotic formulas for eigenvalues of such operators with self-similar weights, expanding understanding of spectral properties in this context.

Findings

Eigenvalues exhibit exponential growth asymptotics.

Asymptotics depend on differential operator order.

Growth rate influenced by self-similarity parameters.

Abstract

The spectral problem for the high order differential operator with singular weight is considered. If the weight is a generalized derivative of self-similar function with zero spectral degree the asymptotics of eigenvalues is obtained. They are proved to have exponential growth and depend on the order of differential operator and self-similarity parameters.