Abstract
In this chapter we will state a result asserting that for elliptic semi-classical (pseudo-)differential operators the eigenvalues are distributed according to Weyl’s law “most of the time” in a probabilistic sense. The first three sections are devoted to the formulation of the results and in the last section we give an outline of the proof that will be carried out in Chaps. 16 and 17.
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Notes
- 1.
\(|\alpha |=\| \alpha \|{ }_{\ell ^1}\), \((hD)^\alpha =h^{|\alpha |}D_{x_1}^{\alpha _1}\cdots D_{x_n}^{\alpha _n}\).
- 2.
When viewed as an 2n − 1 form λ on \((\Re p)^{-1}(t)\), it is given by \(\lambda \wedge d\Re p=dx_1\wedge \cdots \wedge dx_n\wedge d\xi _1 \wedge \cdots \wedge d\xi _n =\) the symplectic volume form.
References
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M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. Math. Ann. 342(1), 177–243 (2008). http://arxiv.org/abs/math/0601381
J. Sjöstrand, Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations. Ann. Fac. Sci. Toulouse 18(4), 739–795 (2009). http://arxiv.org/abs/0802.3584
J. Sjöstrand, Eigenvalue distribution for non-self-adjoint operators on compact manifolds with small multiplicative random perturbations. Ann. Fac. Sci. Toulouse 19(2), 277–301 (2010). http://arxiv.org/abs/0809.4182
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Sjöstrand, J. (2019). Distribution of Eigenvalues for Semi-classical Elliptic Operators with Small Random Perturbations, Results and Outline. 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_15
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DOI: https://doi.org/10.1007/978-3-030-10819-9_15
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