On the Fundamental Eigenvalues gap of Sturm-Liouville Operators

TL;DR

This paper investigates the minimal fundamental eigenvalue gap of Sturm-Liouville operators with specific potential and weight function constraints, using direct optimization methods to identify minimizers.

Contribution

It introduces a novel application of direct optimization techniques to determine eigenvalue gap minimizers under single-well and single-barrier constraints.

Findings

Identified potential and weight configurations that minimize the eigenvalue gap.

Demonstrated the effectiveness of direct optimization in spectral problems.

Provided insights into the structure of extremal Sturm-Liouville operators.

Abstract

We use methods of direct optimization as in [9] to find the minimizers of the fundamental gap of Sturm-Liouville operators on an interval, under the constraint that the potential is of single-well form and that the weight function is of single-barrier form, and under similar constraints expressed in terms of convexity.