Oscillation estimates of eigenfunctions via the combinatorics of noncrossing partitions

TL;DR

This paper establishes bounds on the oscillations of eigenfunctions of a fractional Schrödinger operator by leveraging combinatorial properties of noncrossing partitions, with applications to periodic and Steklov problems.

Contribution

It introduces a novel approach combining Courant's theorem with noncrossing partitions to estimate eigenfunction oscillations for fractional operators.

Findings

Bound on the number of nodal domains for eigenfunction extensions

Estimate of sign changes in eigenfunctions using combinatorics

Applications to periodic and Steklov problems

Abstract

We study oscillations in the eigenfunctions for a fractional Schr\"odinger operator on the real line. An argument in the spirit of Courant's nodal domain theorem applies to an associated local problem in the upper half plane and provides a bound on the number of nodal domains for the extensions of the eigenfunctions. Using the combinatorial properties of noncrossing partitions, we turn the nodal domain bound into an estimate for the number of sign changes in the eigenfunctions. We discuss applications in the periodic setting and the Steklov problem on planar domains.