Almost-periodic ground state of the non-self-adjoint Jacobi operator and its applications

TL;DR

This paper investigates the existence of ground states in non-self-adjoint Jacobi operators within almost periodic media, using dynamical systems methods, and applies findings to reaction-diffusion equations and asymptotic analysis.

Contribution

It establishes the existence of ground states for non-self-adjoint Jacobi operators in almost periodic media, including quasi-periodic cases with lower regularity coefficients, and applies results to related differential equations.

Findings

Existence of ground states in almost periodic media.

Guarantee of ground states with lower regularity in quasi-periodic media.

Applications to discrete Fisher-KPP equations and stationary parabolic equations.

Abstract

We study the ground states of the one-dimensional non-self-adjoint Jacobi operators in the almost periodic media by using the method of dynamical systems. We show the existence of the ground state. Particularly, in the quasi-periodic media, we show that the lower regularity of coefficients can guarantee the existence of ground states. Besides that, we give two applications: the first application is to show the existence and uniqueness of the positive steady state of the discrete Fisher-KPP type equation; the second application is to investigate the asymptotic behavior of the discrete stationary parabolic equation with large lower order terms.