Quasi-Modes in Higher Dimension

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

$$\displaystyle \{ a,b\} =a^{\prime }_\xi \cdot b^{\prime }_x-a^{\prime }_x \cdot b^{\prime }_\xi =H_a(b). $$

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.