Simon N. Chandler-Wilde, Ratchanikorn Chonchaiya, Marko Lindner: Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operator

Author(s):

Simon N. Chandler-Wilde
Ratchanikorn Chonchaiya
Marko Lindner

Title:

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

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 6, 2010

Mathematics Subject Classification:

47B80	[ ]
47A10	[ ]
47B36	[ ]

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.

Keywords:

random matrix, spectral theory, Jacobi matrix, disordered systems

Language:

English

Publication time:

04/2010