Finite sections of the Fibonacci Hamiltonian

TL;DR

This paper proves that finite principal submatrices of the Fibonacci Hamiltonian are stable and invertible for large sizes, with their inverses converging to the inverse of the infinite operator, regardless of truncation method.

Contribution

It establishes the stability and invertibility of finite Fibonacci Hamiltonian submatrices and their convergence to the infinite inverse, regardless of truncation points.

Findings

Finite submatrices are always stable for large sizes.

Invertibility of submatrices is guaranteed for sufficiently large n.

Inverse matrices converge pointwise to the inverse of the infinite operator.

Abstract

We study finite but growing principal square submatrices $A_n$ of the one- or two-sided infinite Fibonacci Hamiltonian $A$. Our results show that such a sequence $(A_n)$, no matter how the points of truncation are chosen, is always stable -- implying that $A_n$ is invertible for sufficiently large $n$ and $A_n^{-1}\to A^{-1}$ pointwise.