On the analogues of Szegő's theorem for ergodic operators

Kirsch, Werner; Pastur, L. А.

doi:10.1070/sm2015v206n01abeh004448

Cited by 17 publications

(24 citation statements)

References 21 publications

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“…(66) This means that A ≃ M M S with the natural isomorphism E α ij [A] ↔ F α ij [A]. Then, comparing (14)-(16) and (64)-(65) (with MS instead of M) we deduce that π in (16) is the isomorphism satisfying (17)-(18). Now, let us prove Theoem 1.3.…”

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confidence: 88%

Matrix representations of multidimensional integral and ergodic operators

Kutsenko

2020

Applied Mathematics and Computation

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We provide a representation of the C * -algebra generated by multidimensional integral operators with piecewise constant kernels and discrete ergodic operators. This representation allows us to find the spectrum and to construct the explicit functional calculus on this algebra. In particular, it can be useful for various applications, since almost all discrete approximations of integral and differential operators belong to this algebra. Some examples are also presented: 1) we construct an explicit functional calculus for extended Fredholm integral operators with piecewise constant kernels, 2) we find a wave function and spectral estimates for 3D discrete Schrödinger equation with planar, guided, local potential defects, and point sources. Some problems of approximation of continuous multi-kernel integral operators by the operators with piecewise constant kernels are also discussed.

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confidence: 88%

Matrix representations of multidimensional integral and ergodic operators

Kutsenko

2020

Applied Mathematics and Computation

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“…In this part of the proof we work with the non-averaged quantities Tr (h(g(H) G )) as long as possible. For a concrete model such as the random Anderson model, and additional (model-specific) assumptions, the pointwise formula (4.27) would be the starting point for an almost sure pointwise or stochastic asymptotic analysis beyond the results from [KP15]. In the second part we then apply Theorem 2.5 to show that the coefficients A m are well-defined for q > 2d and that there exist constants C, C ′ such that…”

Section: Proof Of Theorem 26mentioning

confidence: 99%

“…For M ∈ N Tr (h(W L (a)))) = Recently, subleading-order trace asymptotics as in (1.3)-(1.5) have been studied for Schrödinger operators with non-trivial potential [PS14, KP15, EPS17, PS] that fit in the larger class of ergodic operators. For a, say, Z d -ergodic and selfadjoint operator ω → H ω on L 2 (R d ), a natural generalization for the left-hand side of (1.5) is the trace of the operator h(g(H ω ) [−L,L] d ) for a suitable function g. In [KP15] such trace asymptotics were studied for one-dimensional random and quasiperiodic Schrödinger operators on the lattice. For the random Anderson model and concrete choices of functions g, h the authors showed that the leading order term, which is of order L, obeys a central limit theorem.…”

Section: Introductionmentioning

confidence: 99%

Full Szegő-Type Trace Asymptotics for Ergodic Operators on Large Boxes

Dietlein¹

2018

Commun. Math. Phys.

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We prove full Szegő-type large-box trace asymptotics for selfadjoint Z d -ergodic operators Ω ∋ ω → Hω acting on L 2 (R d ). More precisely, let g be a bounded, compactly supported and real-valued function such that the (averaged) operator kernel of g(Hω) decays sufficiently fast, and let h be a sufficiently smooth compactly supported function. We then prove a full asymptotic expansion of the averaged trace Tr h(g(Hω) [−L,L] d ) in terms of the length-scale L.

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“…In this context, recent literature contains asymptotic formulae for a generalised version of the trace : the operator A is replaced by

a (H)

where

H = - Δ + V

is a Schrödinger operator with a real‐valued potential V and

a : R \to R

is a bounded function, for instance a step function. Here, the focus lies on (random) ergodic potentials in and periodic potentials in .…”

Section: Introductionmentioning

confidence: 99%

A Szegő limit theorem for translation‐invariant operators on polygons

Pfirsch

2019

Mathematische Nachrichten

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We prove Szegő-type trace asymptotics for translation-invariant operators on polygons. More precisely, consider a Fourier multiplier =  *  on 2 ( ℝ 2 ) with a sufficiently decaying, smooth symbol ∶ ℝ 2 → ℂ. Let ⊂ ℝ 2 be the interior of a polygon and, for ≥ 1, define its scaled version ∶= ⋅ . Then we study the spectral asymptotics for the operator = , the spatial restriction of onto : for entire functions ℎ with ℎ(0) = 0 we provide a complete asymptotic expansion of trℎ ( ) as → ∞. These trace asymptotics consist of three terms that reflect the geometry of the polygon. If is replaced by a domain with smooth boundary, a complete asymptotic expansion of the trace has been known for more than 30 years. However, for polygons the formula for the constant order term in the asymptotics is new. In particular, we show that each corner of the polygon produces an extra contribution; as a consequence, the constant order term exhibits an anomaly similar to the heat trace asymptotics for the Dirichlet Laplacian. K E Y W O R D S heat trace anomaly, polygons, Szegő-type trace asymptotics, Wiener-Hopf operators M S C ( 2 0 1 0 ) Primary: 47B35; Secondary: 45M05, 47B10, 58J50 Ω ∶= ⋅ Ω Mathematische Nachrichten. 2019;292:1567-1594.

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On the analogues of Szegő's theorem for ergodic operators

Cited by 17 publications

References 21 publications

Matrix representations of multidimensional integral and ergodic operators

Matrix representations of multidimensional integral and ergodic operators

Full Szegő-Type Trace Asymptotics for Ergodic Operators on Large Boxes

A Szegő limit theorem for translation‐invariant operators on polygons

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