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

Abstract

We study operators defined on a Hilbert space defined by a self-affine Delone set \(\Lambda \) and show that the usual trace of a restriction of the operator to finite-dimensional subspaces satisfies a certain \(\limsup \) law controlled by traces on a certain subalgebra. The asymptotic traces are defined through asymptotic cycles, or \({\mathbb {R}}^d\)-invariant distributions of a dynamical system defined by \(\Lambda \). We use this to refine Shubin’s trace formula for certain self-adjoint operators acting on \(\ell ^2(\Lambda )\) and show that the errors of convergence in Shubin’s formula are given by these traces.

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Acknowledgements

We would like to thank J. Kellendonk for discussing questions related to Shubin’s formula and I. Putnam for discussing traces on \(*\)-algebras. Part of R.T.’s travel funding for this project came from a AMS-Simons Travel Grant.

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Correspondence to Rodrigo Treviño.

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Communicated by Stéphane Nonnenmacher.

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Schmieding, S., Treviño, R. Traces of Random Operators Associated with Self-Affine Delone Sets and Shubin’s Formula. Ann. Henri Poincaré 19, 2575–2597 (2018). https://doi.org/10.1007/s00023-018-0700-8

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