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Published in:
2019 | OriginalPaper | Chapter
9. Quasi-Modes in Higher Dimension
Author : Johannes Sjöstrand
Published in: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations
Publisher: Springer International Publishing
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Abstract
Recall that if a(x, ξ) and b(x, ξ) are two C
1-functions defined on some domain in \({\mathbf {R}}^{2n}_{x,\xi }\), then we can define the Poisson bracket to be the C
0-function on the same domain given by Here \(H_a=a^{\prime }_\xi \cdot \partial _x-a^{\prime }_x\cdot \partial _\xi \) denotes the Hamilton vector field of a. The following result is due to Zworski, who obtained it via a semi-classical reduction from the above mentioned result of Hörmander. A direct proof was given in Dencker et al. and here we give a variant. We will assume some familiarity with symplectic geometry.
$$\displaystyle \{ a,b\} =a^{\prime }_\xi \cdot b^{\prime }_x-a^{\prime }_x \cdot b^{\prime }_\xi =H_a(b). $$