All about Fourier integral operators

In general, for the wave equation \(\Delta u - \frac{1}{c^2}\partial^2_t u = 0\) we may use the ansatz \(u = \psi(x)e^{-i\omega t}\) to obtain the Helmholtz equation \(\Delta\psi + k^2\psi = 0\), where \(\omega = ck\). A general solution for the wave equation is \(u = \int \psi_\omega(x) e^{-i\omega t}d\omega\), which means that the Helmholtz equation picks out a single-frequency solution for the wave equation.

Let's now assume the knowledge that the outgoing solutions of the 2-dimensional wave equation for a fixed frequency \(\omega\) are given, in polar coordinates, by

\begin{align} u(x) = \sum_{n=0}^\infty a_n e^{in\theta - i\omega t}H^1_n(kr), \end{align}

where \(a_n\in\ell^2(\mathbb{N})\) and \(H^1_n(x)\) is the Hankel function of first kind. This is an orthogonal \(\ell^2\) sum, and then the author examines a single term to find the asymptotic for \(k \gg 1\):

\begin{align} k^{\frac{1}{2}}H^1_n(kr) = \sqrt{\frac{2}{\pi r}} e^{ikr - n\pi/2 - \pi/4}\sum_{m=0}^\infty c_m(r)k^{-m} + O(k^nr^ne^{-2kr}). \end{align}

He defines \(H^M_n(kr)\) as the formal truncated sum above at \(m \leq M\), and sets \(u_M(x) \coloneq k^{\frac{1}{2}}H^M_n(kr)e^{in\theta - i\omega t}\) which has the property that \(\square u_M = O(k^{-M-1})\) and \(u_M \to u\), and the author asks then under what conditions can this situation be generalized.

The definition of an asymptotic expansion is given:

\begin{align} b(y,\varepsilon) \simeq \sum b_j(y)\varepsilon^j. \end{align}

The Helmholtz equation is examined again in the high frequency limit resulting in the eikonal and transport equations.