Symmetries of the Feinberg–Zee random hopping matrix

Abstract

We study the symmetries of the spectrum of the Feinberg–Zee Random Hopping Matrix introduced in [J. Feinberg and A. Zee, Spectral curves of non-Hermitian Hamiltonians, Nucl. Phys. B 552 (1999) 599–623] and studied in various papers thereafter (e.g. [S. N. Chandler-Wilde, R. Chonchaiya and M. Lindner, eigenvalue Problem meets Sierpinski triangle: Computing the spectrum of a non-self-adjoint random operator, Oper. Matrices 5 (2011) 633–648; S. N. Chandler-Wilde, R. Chonchaiya and M. Lindner, On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators, Oper. Matrices 7 (2013) 739–775; S. N. Chandler-Wilde and E. B. Davies, Spectrum of a Feinberg–Zee sandom hopping matrix, J. Spectral Theory 2 (2012) 147–179; R. Hagger, On the spectrum and numerical range of tridiagonal random operators, preprint (2014), arXiv: 1407.5486; D. E. Holz, H. Orland and A. Zee, On the remarkable spectrum of a non-Hermitian random matrix model, J. Phys. A: Math. Gen. 36 (2003) 3385–3400]). In [J. Spectral Theory 2 (2012) 147–179], Chandler-Wilde and Davies proved that the spectrum of the Feinberg–Zee Random Hopping Matrix is invariant under taking square roots, which implied that the unit disk is contained in the spectrum (a result already obtained slightly earlier in [Oper. Matrices 5 (2011) 633–648]. In a similar approach we show that there is an infinite sequence of symmetries at least in the periodic part of the spectrum (which is conjectured to be dense). Using these symmetries and the result of [J. Spectral Theory 2 (2012) 147–179], we can exploit a considerably larger part of the spectrum than the unit disk. As a further consequence we find an infinite sequence of Julia sets contained in the spectrum. These facts may serve as a part of an explanation of the seemingly fractal-like behavior of the boundary.