Estimate for an oscillatory integral of the first kind

Question

I am confused in finding the right bound for the following oscillatory integral

$$I = \int_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$

Where $\psi(2^{-k} \xi)$ is a smooth cutt-off function supported on the annulus $A:= \{ 2^{k-1} \leq | \xi| \leq 2^{k+1} \}$, $y \in \mathbb{R}$, $t >0$ and $\eta \in \mathbb{N}$.

The estimate I found using Van der Corput's lemma is as follows:

$$|I| \leq \frac{ 2^{1-2k}}{|\eta \,t|^{\frac{1}{2}}}.$$

Could you please check if my result is correct? I am going to use this result to build many things on it so the last thing I want is building my solution on a not right estimate. That's why I posted it here. Thanks in Advance

Answer 1

Write $s=\eta t$ and note that $I(s,y)$ solves the 1D Schrödinger equation $iI_s-3I_{yy}=0$. Thus it satisfies the sharp estimate $|I(s,y)|\le c_0s^{-1/2}$ where $c_0$ is a multiple of $\int|I(0,y)|dy$. Now, $I(0,y)$ is the Fourier transform of $\chi(2^{-k}\xi)$ where $\chi=\psi^2$, that is to say $I(0,y)=2^k\widehat\chi(2^ky)$, so its $L^1(R)$ norm should be independent of $k$. In other words it seems that the right estimate is $|I(s,y)|\le c_0|t\eta|^{-1/2}$ with $c_0$ a constant.