Let $phiin C_c^infty(mathbb{R})$ be an even function such that $chi_{(-1/2,1/2)}lephile chi_{(-1,1)}$, where $chi_{(a,b)}$ stands for the indicator function of the interval $(a,b)$. For $lambda>0$ consider the oscillatory integral

$$

I(lambda)=int_mathbb{R} phi(x), exp left(ilambda(x+epsilon|x|^{sqrt{2}})right), dx,

$$

with some fixed (very small) positive constant $epsilon$.

My question is: what is the asymptotic behavior of this integral as $lambdarightarrow infty$? I can show, by essentially doing careful integration by parts, that the upper bound is $lesssim lambda^{-sqrt{2}}$, but I wonder whether $lambda^{-sqrt{2}}$ is also a lower bound?

Note, that if the exponent $sqrt{2}$ is replaced by $2k$ for some positive integer $k$, then the integral decays like $lambda^{-M}$ for any $M>0$ due to the non-stationary phase estimate (the derivative of the function $x+epsilon x^{2k}$ is $gtrsim 1$).

I would appreciate any hints on how to approach this problem.