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 [6] and studied in various papers therafter (e.g.[1], [2], [3], [7], [9]). In [3], 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 [1]). 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 [3], 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 behaviour of the boundary.

The eigenvalues of tridiagonal sign matrices are dense in the spectra of periodic tridiagonal sign operators

“…Moreover, it was shown in [5] that π ∞ is dense in the unit disk D := {z ∈ C : |z| ≤ 1}, i.e. clos(π ∞ ) ∩ D = D. This result was improved to a larger set in [10]. Our new result here implies that σ ∞ is dense in this particular set as well.…”

“…Thus the assertion follows. Alternatively one could also apply Laplace's formula twice to get the same result (see [10] for details). 2…”

“…In [9] this upper bound was further improved using second order numerical ranges. Furthermore, π ∞ has an infinite number of symmetries that also carry over to σ ∞ (see [4] for rotations and reflections, [5] for square roots and [10] for higher roots). Combining all these results mentioned above, we have a pretty accurate description of the set σ ∞ .…”

Chebyshev-polynomial expansion of the localization length of Hermitian and non-Hermitian random chains

“…We here use a random-sign model, also known as the Feinberg-Zee (FZ) random-hopping model [27,39,[56][57][58][59][60][61][62][63], defined by the Hamiltonian…”

On Symmetries of the Feinberg–Zee Random Hopping Matrix

“…(1.7) (Here p −2 (S) := p −1 (p −1 (S)), p −3 (S) := p −1 (p −2 (S)), etc.) Hagger [17] observes that, as a consequence of (1.5) and (1.6), U (p) ⊂ Σ π . He also notes that standard results of complex dynamics (e.g., [11,Corollary 14.8]) imply that J(p) ⊂ U (p), so that J(p) ⊂ Σ π ; here J(p) denotes the Julia set of the polynomial p. (Where p 2 (λ) := p(p(λ)), p 3 (λ) := p(p 2 (λ)), etc., we recall [11] that the filled Julia set K(p) of a polynomial p of degree ≥ 2 is the compact set of those λ ∈ C for which the sequence (p n (λ)) n∈N , the orbit of λ, is bounded.…”