Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operator

Chandler-Wilde, Simon N.; Chonchaiya, Ratchanikorn; Lindner, Marko

doi:10.7153/oam-05-46

Cited by 22 publications

(85 citation statements)

References 34 publications

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“…The following theorem confirms the conjecture by Chandler-Wilde, Chonchaiya and Lindner [3,4] that σ ∞ is even dense in π ∞ .…”

Section: Three Lemmas and A Theoremsupporting

confidence: 82%

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

Hagger

2015

Journal of Functional Analysis

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Chandler-Wilde, Chonchaiya and Lindner conjectured that the set of eigenvalues of finite tridiagonal sign matrices (i.e. plus and minus ones on the first sub-and superdiagonal, zeroes everywhere else) is dense in the set of spectra of periodic tridiagonal sign operators on the usual Hilbert space of square summable bi-infinite sequences. We give a simple proof of this conjecture. As a consequence we get that the set of eigenvalues of tridiagonal sign matrices is dense in the unit disk. In fact, a recent paper further improves this result, showing that this set of eigenvalues is dense in an even larger set.

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“…The following theorem confirms the conjecture by Chandler-Wilde, Chonchaiya and Lindner [3,4] that σ ∞ is even dense in π ∞ .…”

Section: Three Lemmas and A Theoremsupporting

confidence: 82%

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

Hagger

2015

Journal of Functional Analysis

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“…Note, however, that Eqs. (29) and (30) always deliver a value for κ(λ), regardless of whether there is actually a normalizable eigenfunction at that particular value of λ.…”

Section: B Transfer Matrix Approachmentioning

confidence: 99%

Non-Hermitian localization in biological networks

2016

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We explore the spectra and localization properties of the N -site banded one-dimensional nonHermitian random matrices that arise naturally in sparse neural networks. Approximately equal numbers of random excitatory and inhibitory connections lead to spatially localized eigenfunctions, and an intricate eigenvalue spectrum in the complex plane that controls the spontaneous activity and induced response. A finite fraction of the eigenvalues condense onto the real or imaginary axes. For large N , the spectrum has remarkable symmetries not only with respect to reflections across the real and imaginary axes, but also with respect to 90• rotations, with an unusual anisotropic divergence in the localization length near the origin. When chains with periodic boundary conditions become directed, with a systematic directional bias superimposed on the randomness, a hole centered on the origin opens up in the density-of-states in the complex plane. All states are extended on the rim of this hole, while the localized eigenvalues outside the hole are unchanged. The bias dependent shape of this hole tracks the bias independent contours of constant localization length. We treat the large-N limit by a combination of direct numerical diagonalization and using transfer matrices, an approach that allows us to exploit an electrostatic analogy connecting the "charges" embodied in the eigenvalue distribution with the contours of constant localization length. We show that similar results are obtained for more realistic neural networks that obey "Dale's Law" (each site is purely excitatory or inhibitory), and conclude with perturbation theory results that describe the limit of large bias g, when all states are extended. Related problems arise in random ecological networks and in chains of artificial cells with randomly coupled gene expression patterns.

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“…where the rectangle marks the matrix entry at (0, 0). When A is the set {−1, 1}, the corresponding operator A b is related to the so called "hopping sign model" introduced in [7] and subsequently studied in many other works, such as [1,[3][4][5][9][10][11], just to name a few.…”

Section: Introductionmentioning

confidence: 99%

The numerical range of a class of periodic tridiagonal operators

Itzá-Ortiz

Martínez-Avendaño

2020

Linear and Multilinear Algebra

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In this paper we compute the closure of the numerical range of certain periodic tridiagonal operators. This is achieved by showing that the closure of the numerical range of such operators can be expressed as the closure of the convex hull of the uncountable union of numerical ranges of certain symbol matrices. For a special case, this result can be improved so that it is the convex hull of the union of the numerical ranges of only two matrices. A conjecture is stated for the general case.

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Eigenvalue problem meets Sierpinski triangle: computing the spectrum of a non-self-adjoint random operator

Cited by 22 publications

References 34 publications

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

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

Non-Hermitian localization in biological networks

The numerical range of a class of periodic tridiagonal operators

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