Let
- $f : \mathbb{R} \to \mathbb{R}$ a smooth function of exponential decay at infinity (I think of something like a Gaussian) ;
- $g$ a polynomial of $\mathbb{R}$.
I would like to have a quantitative version of the following statement :
'' The Lagragian state $$ \Psi_{f,g}^h := y \mapsto f(y) \ e^{\frac{ i \pi g(y)}{h}} $$ microlocalises near the graph $\Gamma(g')$ of $g'$ on $T \mathbb{R} = \mathbb{R}^2$. ''
More precisely, let $(q,p) \in \mathbb{R}^2$ (an element of $T \mathbb{R} = \mathbb{R}^2$) and $h > 0$ a parameter. We recall the definition of a wave packet that microlocalises toward $(q,p)$ as $h \to 0$,
$$ \Phi_{(q,p)}^h(y) := C(h) \ e^{\frac{2i \pi p y}{h}} \ e^{\frac{- \pi (y-q)^2}{h}} \ , $$ where $C(h)$ is a constant such that $|| \Phi_{(q,p)}^h ||_2 = 1$.
The question is :
Is there a function $\tilde{f} : \mathbb{R} \to \mathbb{R}$ with exponential decay at infinity and two constants $C_1, C_2 > 0$ such that for any $h \le 1$ we have $$ \Big| \langle \Phi_{(q,p)}^h \cdot \Psi_{f,g}^h \rangle_{L^2} \Big| \le \frac{\tilde{f}(q)}{h^{C_1}} \cdot \exp{ \left( { \frac{- C_2 \cdot d( \Gamma(g'), (q,p))^2}{h} } \right)} \ , $$
where $d( \Gamma(g'), (q,p))$ is the distance for the euclidean metric on $\mathbb{R}^2$ from the graph $\Gamma(g')$ of $g'$ to the point $(q,p)$ ?
Some preliminary explicit computations show me that it should be the case but I cannot figure out a general argument.
