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

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