Asymptotic behavior of the curves in the Fu{\v{c}}{\'{\i}}k spectrum

TL;DR

This paper investigates the asymptotic properties of the Fu{c}ik spectrum curves for weighted second order linear ODEs, establishing Weyl-type asymptotics and providing an algorithm for spectrum intersection computation.

Contribution

It introduces a Weyl-type asymptotic analysis for the Fu{c}ik spectrum and presents an algorithm to compute spectrum intersections with rays, linking asymptotics to weights.

Findings

Weyl-type asymptotics for spectrum curves derived

Algorithm for spectrum intersection computation developed

Comparison between computed and asymptotic spectrum values conducted

Abstract

In this work we study the asymptotic behavior of the curves of the Fu{\v{c}}{\'{\i}}k spectrum for weighted second order linear ordinary differential equations. We prove a Weyl type asymptotic behavior of the hyperbolic type curves in the spectrum in terms of some integrals of the weights. We present an algorithm which computes the intersection of the Fu{\v{c}}{\'{\i}}k spectrum with rays through the origin, and we compare their values with the asymptotic ones.