Abstract
We prove full Szegő-type large-box trace asymptotics for selfadjoint \({\mathbb{Z}^d}\)-ergodic operators \({\Omega\ni \omega\mapsto H_\omega}\) acting on \({L^2(\mathbb{R}^d)}\). More precisely, let g be a bounded, compactly supported and real-valued function such that the (averaged) operator kernel of \({g(H_\omega)}\) decays sufficiently fast, and let h be a sufficiently smooth compactly supported function. We then prove a full asymptotic expansion of the averaged trace \({{\rm Tr}\left( h(g(H_\omega)_{[-L,L]^d}) \right)}\) in terms of the length-scale L.
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The author is very grateful to Peter Müller, Bernhard Pfirsch and Alexander Sobolev for many illuminating discussions on this topic.
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Communicated by L. Erdős
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Dietlein, A. Full Szegő-Type Trace Asymptotics for Ergodic Operators on Large Boxes. Commun. Math. Phys. 362, 983–1005 (2018). https://doi.org/10.1007/s00220-018-3161-5
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DOI: https://doi.org/10.1007/s00220-018-3161-5