Abstract
In this chapter we consider elliptic differential operators on a compact manifold and rather than taking the semi-classical limit (h →), we let h = 1 and study the distribution of large eigenvalues. Bordeaux Montrieux (Loi de Weyl presque sûre et résolvante pour des opérateurs différentiels non-autoadjoints, thèse, CMLS, Ecole Polytechnique, 2008. https://pastel.archives-ouvertes.fr/pastel-00005367, Ann Henri Poincaré 12:173–204, 2011) studied elliptic systems of differential operators on S 1 with random perturbations of the coefficients, and under some additional assumptions, he showed that the large eigenvalues obey the Weyl law almost surely. His analysis was based on a reduction to the semi-classical case, where he could use and extend the methods of Hager (Ann Henri Poincaré 7:1035–1064, 2006).
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References
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Sjöstrand, J. (2019). Distribution of Large Eigenvalues for Elliptic Operators. In: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations. Pseudo-Differential Operators, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10819-9_18
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