Spectral properties of integral operators in bounded, large intervals

TL;DR

This paper analyzes the spectral properties of integral operators in large bounded intervals, revealing an exponentially vanishing minimal eigenvalue and a uniform spectral gap, with applications to nonlocal phase transition models.

Contribution

It provides a Perron-Frobenius theorem for the spectrum and bounds the spectral gap uniformly in large intervals, advancing understanding of nonlocal operator spectra.

Findings

Minimal eigenvalue is positive and decreases exponentially with interval size.

Spectral gap is uniformly bounded below using a generalized Cheeger's inequality.

Results facilitate analysis of nonlocal Cahn-Hilliard equations in interface dynamics.

Abstract

We study the spectrum of one dimensional integral operators in bounded real intervals of length $2L$, for value of $L$ large. The integral operators are obtained by linearizing a non local evolution equation for a non conserved order parameter describing the phases of a fluid. We prove a Perron-Frobenius theorem showing that there is an isolated, simple minimal eigenvalue strictly positive for $L$ finite, going to zero exponentially fast in $L$. We lower bound, uniformly on $L$, the spectral gap by applying a generalization of the Cheeger's inequality. These results are usefulfor deriving spectral properties for non local Cahn-Hilliard type of equations in problems of interface dynamics.