Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operator

TL;DR

This paper proves that the spectrum of a specific non-self-adjoint random operator includes interior points by linking it to the eigenvalues of matrices structured like a Sierpinski triangle.

Contribution

It establishes that the spectrum of a particular non-self-adjoint operator contains interior points, using a novel construction involving the Sierpinski triangle pattern.

Findings

The spectrum has interior points.

Constructed an infinite matrix with Sierpinski pattern.

Every point in the open unit disk is an eigenvalue.

Abstract

The purpose of this paper is to prove that the spectrum of the non-self-adjoint one-particle Hamiltonian proposed by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433--6443) has interior points. We do this by first recalling that the spectrum of this random operator is the union of the set of $\ell^\infty$ eigenvalues of all infinite matrices with the same structure. We then construct an infinite matrix of this structure for which every point of the open unit disk is an $\ell^\infty$ eigenvalue, this following from the fact that the components of the eigenvector are polynomials in the spectral parameter whose non-zero coefficients are $\pm 1$'s, forming the pattern of an infinite discrete Sierpinski triangle.