Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula

Scott Schmieding; Rodrigo Treviño

doi:10.1007/s00023-018-0700-8

Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula

Scott Schmieding, Rodrigo Treviño

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We study operators defined on a Hilbert space defined by a self-affine Delone set Λ and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain lim sup law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or R^d-invariant distributions of a dynamical system defined by Λ. We use this to refine Shubin’s trace formula for certain self-adjoint operators acting on ℓ²(Λ) and show that the errors of convergence in Shubin’s formula are given by these traces.

Original language	English (US)
Pages (from-to)	2575-2597
Number of pages	23
Journal	Annales Henri Poincare
Volume	19
Issue number	9
DOIs	https://doi.org/10.1007/s00023-018-0700-8
State	Published - Sep 1 2018

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
Nuclear and High Energy Physics
Mathematical Physics

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10.1007/s00023-018-0700-8

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abstract = "We study operators defined on a Hilbert space defined by a self-affine Delone set Λ and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain lim sup law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or Rd-invariant distributions of a dynamical system defined by Λ. We use this to refine Shubin{\textquoteright}s trace formula for certain self-adjoint operators acting on ℓ2(Λ) and show that the errors of convergence in Shubin{\textquoteright}s formula are given by these traces.",

author = "Scott Schmieding and Rodrigo Trevi{\~n}o",

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AU - Treviño, Rodrigo

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N2 - We study operators defined on a Hilbert space defined by a self-affine Delone set Λ and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain lim sup law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or Rd-invariant distributions of a dynamical system defined by Λ. We use this to refine Shubin’s trace formula for certain self-adjoint operators acting on ℓ2(Λ) and show that the errors of convergence in Shubin’s formula are given by these traces.

AB - We study operators defined on a Hilbert space defined by a self-affine Delone set Λ and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain lim sup law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or Rd-invariant distributions of a dynamical system defined by Λ. We use this to refine Shubin’s trace formula for certain self-adjoint operators acting on ℓ2(Λ) and show that the errors of convergence in Shubin’s formula are given by these traces.

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ER -

Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula

Abstract

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