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
