Abstract
For the Toeplitz quantization of complex-valued functions on a 2n-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law (1.3). In numerical experiments the same Weyl law also holds for “false” eigenvalues created by pseudospectral effects.
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Acknowledgements
We would like to thank Edward Bierstone and Pierre Milman for helpful discussions of Łojasiewicz inequalities, Mark Rudelson for suggestions concerning random matrices, and Stéphane Non-nenmacher and Michael Van Valkenburgh for comments on early versions of the paper. The authors gratefully acknowledge the partial support by an MU research leave, and NSF grants DMS 0500267, DMS 0654436. The first author thanks the Mathematics Department of U.C. Berkeley for its hospitality in spring 2009.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Christiansen, T.J., Zworski, M. Probabilistic Weyl Laws for Quantized Tori. Commun. Math. Phys. 299, 305–334 (2010). https://doi.org/10.1007/s00220-010-1047-2
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DOI: https://doi.org/10.1007/s00220-010-1047-2