TITLE: Is there something wrong with Hörmander's theorem on stationary phase method
QUESTION [10 upvotes]: It is well-know that the Bessel function has the asymptotic expansion $J_n(\omega) \sim \left( \frac 2 {\pi \omega} \right)^{1/2} \left( \cos \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) - \sin \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) \frac {4n^2 - 1} {8 \omega} ...\right)$.
We also have the integral representation $$J_n(\omega) = \int_0^1 \cos (n \pi x - \omega \sin \pi x) dx = \Re  \int_0^1 e^{n \pi i x} e^{ - \omega \sin \pi x } dx.$$
Now we look at Hörmander's Theorem 7.7.5(The Analysis of Linear Partial Differential Operators I). The phase function is $f(x) = - \sin \pi x $, and has stationary point $x_0 = 1/2$. The weight function is $u(x) = e^{n \pi i x} $.  Hörmander's theorem says \begin{equation}
\begin{split}
\int_0^1 u(x) e^{i \omega f(x)} d x & \sim e^{i \omega f(1/2)} (\det (\omega f''(1/2)/ 2 \pi i))^{-1/2} (u (1/2) + L_1 u \omega^{-1} )\\
& = \left( \frac 2 {\pi x} \right)^{1/2} e^{- i \omega + \pi i /4}  (e^{n \pi i /2} + L_1 u \omega^{-1} ).
\end{split}
\end{equation}
The first term is right. Let us look at the second term.
We have $f(1/2) = -1$, $f''(1/2) = \pi^2$, so $g_{1/2} (x) = - \sin \pi x + 1 - (\pi^2/2) (x-1/2)^2$. We note that $g_{1/2} (x)$ vanishes not only of third order but of fourth order at $1/2$, that is, $g(1/2) = g'(1/2) = g''(1/2) = g'''(1/2) = 0$, and $g''''(1/2) = - \pi^4$.  $L_1 u $ is defined as
$$i^{-1} \sum_{\nu - \mu = 1} \sum_{2\nu \geq 3 \mu} 2^{-\nu} \langle \pi^{-2} D, D \rangle^{\nu} (g_{1/2}^{\mu} u) (1/2) /\mu! \nu! .$$
We see that 
$$L_1 u = i^{-1} \left( 2^{-1} \pi^{-2} (n \pi i)^2 e^{n \pi i /2} + 2^{-3} \pi^{-4} (- \pi^4 )  e^{n \pi i /2} + 0\right) = i e^{n \pi i /2} \left( \frac 1 2 n ^2 + \frac 1 8 \right).$$
Therefore the second term is $ \sin \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) \frac {4n^2 + 1} {8 \omega}$ according to Hömander instead of $ - \sin \left(\omega -\frac 1 2 n \pi - \frac 1 4 \pi\right) \frac {4n^2 - 1} {8 \omega}$. My guess is that there is a factor $(-1)^{\nu}$ missing in Hörmander's theorem, then also in  Hörmander's Lemma 7.7.3., but this lemma looks right.
Could anyone tell me what goes wrong here? I would really appreciate it!

REPLY [9 votes]: Looking closely again, I am pretty sure my comment gives the answer. Note that Hörmander uses the notation $D = - i \partial$. So in particular
$$ Du = -i (n\pi i) u $$
when $u = e^{n\pi i x}$. This gives an extra minus sign in the first term ($\nu = 1$ and $\mu = 0$) of the definition for $L_1 u$, while leaving the second term ($\nu = 2$ and $\mu = 1$) unchanged, providing the extra $(-1)^\nu$ you are looking for.
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