Approximating spectral invariants of Harper operators on graphs II

Mathai, Varghese; Schick, Thomas; Yates, Stuart

doi:10.1090/s0002-9939-02-06739-4

Cited by 14 publications

(2 citation statements)

References 5 publications

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“…The proof of this theorem in the case of H = 1 was given in [18] and [19] for the discrete magnetic Laplacian. To establish this theorem in our more general situation, we apply the same arguments, slightly generalized as follows, relying upon the notation established in the proof of Theorem 6.2.…”

Section: Generalized Integrated Density Of States and Spectral Gapsmentioning

confidence: 99%

Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs

Dodziuk

Mathai

Yates

2005

Commun. Math. Phys.

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Let Q denote the field of algebraic numbers in C. A discrete group G is said to have the σ-multiplier algebraic eigenvalue property, if for every matrix A ∈ M d (Q(G, σ)), regarded as an operator on l 2 (G) d , the eigenvalues of A are algebraic numbers, where σ ∈ Z 2 (G, U(Q)) is an algebraic multiplier, and U(Q) denotes the unitary elements of Q. Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier σ. In the special case when σ is rational (σ n =1 for some positive integer n) this property holds for a larger class of groups K containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators.2000 Mathematics Subject Classification. Primary: 58G25, 39A12.

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Section: Generalized Integrated Density Of States and Spectral Gapsmentioning

confidence: 99%

Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs

Dodziuk

Mathai

Yates

2005

Commun. Math. Phys.

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“…[1,14,31] for the continuous setting and e.g. [4,23,24] for discrete geometries. It is well known that weak convergence of distribution functions only implies pointwise convergence at the continuity points of the limit function.…”

Section: Introductionmentioning

confidence: 99%

Uniform Existence of the Integrated Density of States on Metric Cayley Graphs

2013

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Given an arbitrary, finitely generated, amenable group we consider ergodic Schrödinger operators on a metric Cayley graph with random potentials and random boundary conditions. We show that the normalised eigenvalue counting functions of finite volume parts converge uniformly. The integrated density of states (IDS) as the limit can be expressed by a Pastur-Shubin formula. The spectrum supports the corresponding measure and discontinuities correspond to the existence of compactly supported eigenfunctions. In this context, the present work generalises the hitherto known uniform IDS-approximation results for operators on the d-dimensional metric lattice to a very large class of geometries.MSC 2010: 47E05, 34L40, 47B80, 81Q10

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Approximation of the Integrated Density of States on Sofic Groups

Schumacher

Schwarzenberger

2014

Ann. Henri Poincaré

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In this paper we study spectral properties of self-adjoint operators on a large class of geometries given via sofic groups. We prove that the associated integrated densities of states can be approximated via finite volume analogues. This is investigated in the deterministic as well as in the random setting. In both cases we cover a wide range of operators including in particular unbounded ones. The large generality of our setting allows to treat applications from long-range percolation and the Anderson model. Our results apply to operators on Z d , amenable groups, residually finite groups and therefore in particular to operators on trees. All convergence results are established without any ergodic theorem at hand.

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Approximating spectral invariants of Harper operators on graphs II

Cited by 14 publications

References 5 publications

Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs

Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs

Uniform Existence of the Integrated Density of States on Metric Cayley Graphs

Approximation of the Integrated Density of States on Sofic Groups

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