1

Let $\phi(x) = a(x)\exp\left({i \cfrac{\xi_0.x}{h}}\right) $, with $\xi_0 \in \mathbb R^d$ fixed and $a \in C^{\infty}_c(\mathbb R^d)$. Let $f \in C^{\infty}_c(\mathbb R^{2d}) $. I want to show that $$ (Op^W_h(f)\phi)(x) = b(x,h)\exp\left({i \cfrac{\xi_0.x}{h}}\right)$$ with $b$ verifying an asymptotic expansion, $$ b(x,h) \sim \sum_{j=0}^{\infty}h^jb_j(x),$$ and explicitly find $b_0$ and $b_1$ ($Op^W_h$ being the Weyl quantization). The exercise says to use the quadratic stationnary phase lemma, but I don’t know how to do it. Let me precise what we have here.

$$ (Op^W_h(f)\phi)(x) = (2\pi h)^{-d} \int \int \exp\left({i \cfrac{(x-y).\eta}{h}}\right) f\left(\cfrac{x-y}{2},\eta\right)\phi(y)dyd\eta,$$ so we get

$$ (Op^W_h(f)\phi)(x) = (2\pi h)^{-d} \int \int \exp\left({i \cfrac{(x-y).\eta}{h}}\right) f\left(\cfrac{x-y}{2},\eta\right)a(y)\exp\left({i \cfrac{\xi_0.y}{h}}\right)dyd\eta.$$

Could someone please explain at what element in this double integral I should apply the stationary phase lemma, which is for integrals looking like $$\int c(x)\exp\left({i \cfrac{\psi(x)}{h}}\right)dx $$ and why « quadratic » ? Any suggestion would be nice, thanks.

zarathustra
  • 191
  • Could you be more precise about what you want to show? – kieransquared Jan 04 '23 at 23:14
  • @kieransquared I want to show that $(Op^W_h(f)\phi)$ taking $x$ as an argument can be written as $b(x,h) \exp\left({i \cfrac{\xi_0.x}{h}}\right) $, with $b$ a certain function that admits an asymptotic formula $ b(x,h) \sim \sum_{j=0}^{\infty}h^jb_j(x)$, thanks to the stationary phase lemma. I have precised to what $(Op^W_h(f)\phi) $ is equal, as a double integral. The lemma deals with $\int c(x)\exp\left({i \cfrac{\psi(x)}{h}}\right)dx $, but I do not find the function $\psi$ that sould be in an the argument of an exponential here, that has a critical point, in order to apply the lemma. – zarathustra Jan 05 '23 at 10:40
  • 1
    I’m asking for clarification on the form of the asymptotic expansion you’re looking for. Are the $h_j$ constants, and what are they allowed to depend on? More generally, it might be helpful to not view the stationary phase as a black box lemma to apply, but instead to use the key techniques (repeated integration by parts, making use of non-degeneracy in the Hessian or Jacobian) to obtain a similar result adapted to your situation. – kieransquared Jan 05 '23 at 13:29

0 Answers0