### 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 ℓ^{2}(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 | |

State | Published - Sep 1 2018 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Nuclear and High Energy Physics
- Mathematical Physics

### Cite this

*Annales Henri Poincare*,

*19*(9), 2575-2597. https://doi.org/10.1007/s00023-018-0700-8

}

*Annales Henri Poincare*, vol. 19, no. 9, pp. 2575-2597. https://doi.org/10.1007/s00023-018-0700-8

**Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula.** / Schmieding, Scott Edward; Treviño, Rodrigo.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula

AU - Schmieding, Scott Edward

AU - Treviño, Rodrigo

PY - 2018/9/1

Y1 - 2018/9/1

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.

UR - http://www.scopus.com/inward/record.url?scp=85049101119&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049101119&partnerID=8YFLogxK

U2 - 10.1007/s00023-018-0700-8

DO - 10.1007/s00023-018-0700-8

M3 - Article

VL - 19

SP - 2575

EP - 2597

JO - Annales Henri Poincare

JF - Annales Henri Poincare

SN - 1424-0637

IS - 9

ER -