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

Hagger, Raffael

doi:10.1016/j.jfa.2015.01.019

Cited by 16 publications

(23 citation statements)

References 10 publications

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“…While many studies [39,[56][57][58][59][60][61][62][63] had focused on the fractallike spectrum of the FZ random-hopping model (67), the study in Ref. [27] found two new features of the model's inverse localization length, which we here reproduce by means of our Chebyshev-polynomial expansion (66).…”

Section: Demonstrationmentioning

confidence: 68%

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

Hatano

Feinberg

2016

Phys. Rev. E

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We study Chebyshev-polynomial expansion of the inverse localization length of Hermitian and non-Hermitian random chains as a function of energy. For Hermitian models, the expansion produces this energy-dependent function numerically in one run of the algorithm. This is in strong contrast to the standard transfer-matrix method, which produces the inverse localization length for a fixed energy in each run. For non-Hermitian models, as in the transfer-matrix method, our algorithm computes the inverse localization length for a fixed (complex) energy. We also find a formula of the Chebyshev-polynomial expansion of the density of states of non-Hermitian models. As explained in detail, our algorithm for non-Hermitian models may be the only available efficient algorithm for finding the density of states of models with interactions.

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Section: Demonstrationmentioning

confidence: 68%

“…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…”

Section: Demonstrationmentioning

confidence: 99%

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

Hatano

Feinberg

2016

Phys. Rev. E

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“…We conjecture that when X b is of finite type the union of the spectrums of the tridiagonal operators corresponding to periodic sequences in X b is dense in the spectrum of A b . For the case of pseudoergodic sequences this was conjectured in [2] and proved recently in [6].…”

Section: Tridiagonal Operators Associated To Subshiftsmentioning

confidence: 78%

A class of tridiagonal operators associated to some subshifts

Hernández-Becerra

Itzá-Ortiz

2016

Open Mathematics

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: We consider a class of tridiagonal operators induced by not necessary pseudoergodic biinfinite sequences. Using only elementary techniques we prove that the numerical range of such operators is contained in the convex hull of the union of the numerical ranges of the operators corresponding to the constant biinfinite sequences; whilst the other inclusion is shown to hold when the constant sequences belong to the subshift generated by the given biinfinite sequence. Applying recent results by S. N. Chandler-Wilde et al. and R. Hagger, which rely on limit operator techniques, we are able to provide more general results although the closure of the numerical range needs to be taken.

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“…For n ∈ N, k ∈ {±1} n , and ϕ ∈ R, let a k (ϕ) := A (n) k + e −iϕ R n + k n e iϕ R T n . The following characterisation of the spectra of periodic operators is well-known (see Lemma 1 and the discussion in [16]).…”

Section: Preliminaries and Previous Workmentioning

confidence: 99%

“…Then, connecting spectra of finite and infinite matrices, it has been shown in [4] that σ n ⊂ π 2n+2 , for n ∈ N, so that σ ∞ ⊂ π ∞ ⊂ Σ π . Further, [16] shows that σ ∞ is dense in π ∞ , so that Σ π = σ ∞ . In Section 3.1 we build on and extend these results, making a surprising connection between the eigenvalues of the finite matrices (1.8) and the spectra of the periodic operators associated to the polynomials in S. The result we prove (Theorem 3.8), is key to the later arguments in Section 4.…”

Section: Introductionmentioning

confidence: 98%

On Symmetries of the Feinberg–Zee Random Hopping Matrix

Chandler-Wilde¹,

Hagger²

2017

Recent Trends in Operator Theory and Partial Differential Equations

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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.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.

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

Cited by 16 publications

References 10 publications

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

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

A class of tridiagonal operators associated to some subshifts

On Symmetries of the Feinberg–Zee Random Hopping Matrix

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