Spectral partitions for Sturm-Liouville problems

TL;DR

This paper investigates optimal interval partitions for Sturm-Liouville problems by analyzing eigenvalue-based functionals, using 3-convergence to understand asymptotic behavior, with applications to spectral sums and heat trace approximations.

Contribution

It introduces a 3-convergence framework for asymptotic analysis of spectral partitions in Sturm-Liouville problems, extending understanding of eigenvalue-based optimization.

Findings

Asymptotic distribution of minimizers characterized

Framework applies to spectral sums and heat trace approximations

Provides insights into optimal spectral interval partitions

Abstract

We look for best partitions of the unit interval that minimize certain functionals defined in terms of the eigenvalues of Sturm-Liouville problems. Via \Gamma-convergence theory, we study the asymptotic distribution of the minimizers as the number of intervals of the partition tends to infinity. Then we discuss several examples that fit in our framework, such as the sum of (positive and negative) powers of the eigenvalues and an approximation of the trace of the heat Sturm-Liouville operator.