WKB in dimension ≥ 2. 2

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\const}{\operatorname{const}}$ $\newcommand{\sgn}{\operatorname{sgn}}$

Global theory

  1. Global construction: phase
  2. Global construction: amplitude
  3. Simple caustic points

Global construction

We considered equation \begin{equation} -ih^{-1}u_t + H(x,hD_x,h) u=0. \label{eq-7.2.1} \end{equation} We constructed solution \begin{equation} u_h(x,t)= e^{ih^{-1}S(x,t)} A(x,t,h) \label{eq-7.2.2} \end{equation} with \begin{equation} A(x,t,h)\sim \sum_{n\ge 0} A_n (x,t)h^n \label{eq-7.2.3} \end{equation} before it hits points where $S(x,t)$ is no more a smooth function of $x$. However smooth Lagrangian manifold $\Lambda_t$ is constructed globally and $S(z,t)$ is a smooth function of $z\in \Lambda_t$.

Near points where $\pi_x:\Lambda_t\ni (x,p)\to x$ is no more local diffeomorphism we find such a partition $(I,J)$ of $\{1,\ldots,n\}$ that $\pi_I:\Lambda \ni (x,p)\to (x_I, p_J)$ is a local diffeomorphism (see Lemma 7.1.3). Then we make a partial Fourier transform (see Definition 7.1.2). Then according to Theorem 7.1.1 \begin{equation} (F_Ju)(x_I,p_J,t) = e^{-ih^{-1}\tilde{S}(x_I,p_J,t)} \tilde{A}(x_J,P_J,h) \label{eq-7.2.4} \end{equation} where $\tilde{S}(p)$ is a partial Legendre transform of $S(x)$: \begin{equation} \tilde{S}(p)= p_J\cdot x_J(x_I,p_J) - S(x_I, x_J) \label{eq-7.2.5} \end{equation} where $x_J=x_J(x_I,p_J)$ is defined from $S\nabla_{x_J} =p_J$, and $\tilde{A}(x_I,p_J,h)\sim \sum_n \tilde{A}_n(x_I,p_J) h^n$ with \begin{equation} \tilde{A}_0(p)= \frac{1}{\sqrt{|S_{x_Jx_J}|}} e^{-\frac{i\pi}{4}\sgn(S_{x_Jx_J})} A(x_I,x_J). \label{eq-7.2.6} \end{equation}

Further, it solves (\ref{7.2.1}) in $(x_I,p_J)$-representation \begin{equation} -ih^{-1}v_t + H(x_I,-hD_{p_J},hD_{x_I},p_J,h) v=0 \label{eq-7.2.7} \end{equation} and therefore $\tilde{S}$ satisfies corresponding Hamilton-Jacobi equation \begin{equation} -\tilde{S}_t + H_0(x_I,\tilde{S}_{p_J}, -\tilde{S}_{x_I},p_J)=0 \label{eq-7.2.8} \end{equation} and $\tilde{A}_n$ satisfy corresponding transport equations and all those equations work as long as projection $\pi_I:\Lambda_t\ni (x,p)\to (x_I,p_J)$ is remains local diffeomorphism.

Furthermore, after $\pi_x:\Lambda_t\ni (x,p)\to x$ is again local diffeomorphism we can make inverse transform \begin{equation} u_h (x,t) = F_J^{-1}v = (2\pi h)^{-\frac{d-m}{2}} \ \int e^{ih^{-1} p_J\cdot x_J } v_h(x_I,p_J,t)\,dp \label{eq-7.2.9} \end{equation} and again get solution (\ref{eq-7.2.2}) with $S(z,t)$ already constructed globally and $\tilde{S}(z,t)$ too.

Then we can continue until $\pi_x$ is no more local diffeomorphism and so on.

Global construction: amplitude

What about amplitudes? We are looking mainly for a leading term $A_0(x,t)$. First, it has singularities as $\pi_x$ is no longer a local diffeomorphism. These singularities of $A_0(x,t)$ and of $\tilde{A}_0(p,t)$ could be "tamed" if we consider \begin{equation} a_0(z,t)= A_0(z,t)|\frac{dx}{dz}|^{\frac{1}{2}} \label{eq-7.2.10} \end{equation} and \begin{equation} \tilde{a_0}(z,t)= \tilde{A}_0(z,t)|\frac{dx}{dz}|^{\frac{1}{2}} \label{eq-7.2.11} \end{equation} where $dz$ is a measure on $\Lambda_t$ which is invariant with respect to Hamiltonian flow (so we take original $dz=dx$ as $t=0$ and push it forward).

Definition 1. We say that $a_0(z,.)$ is half-density because $|a_0(z,.)|^2$ is a density i.e. $|a_0(z,.)|^2\,dz$ does not change as we change $dz$.

Further, since $S_{x_Jx_J}=\tilde{S}_{p_Jp_J}^{-1}$ at points where both $\pi_x$ and $\pi_I$ are local diffeomorphisms, $a_0(x,t)$ and $\tilde{a}_0(x_I, p_J(x),t)$ almost coincide. What is the difference? Factor \begin{equation} e^{-\frac{i\pi}{4}\sgn(\frac{dp}{dx})} \label{eq-7.2.12} \end{equation} means that $a_0$ acquires factor $1=e^{-\frac{i}{2}\eta(z^*)}$ with $\eta(z^*)=\sgn (S_{xx}(z^-))-\sgn(S_{xx}(z^+))$ where $z^\mp$ is a point before/after $z^*$.

Therefore we arrive to

Definition 2. Consider a path $\gamma$ in which there are several points $z^*_k$, $k=1,\ldots, N$ in which $\rank (d\pi_{xx})< d$ and $\rank (d\pi_{xx})= d in all other points. Then $\iota_M(\gamma)$ is called Maslov index of $\gamma$.

Remark 1.

  1. We can define Maslov index without $t$, just on a single manifld $\Lambda$.
  2. We are interested in Maslov index modulo $4$ since $i=e^{-\frac{i}{2}n}=1$ as $n\equiv 0 \mod 4$.
  3. We are especially interested in Maslov index of the closed path $\gamma$. In this case Maslov index does not depend on the choice of the start point (which is also end point) of $\gamma$ but depends on orientation. However Maslov index $\mod 4$ does not depend on orientation.
  4. For closed path Maslov index $\mod 4$ does not change if we permute $x$ and $p$.

Simple caustic points

Consider caustic point $(\bar{z})$ \begin{equation} \rank (d\pi_x)(z^*)< d \label{eq-7.2.13} \end{equation} and assume that it is simple i.e. satisfies (\ref{7.2.14}) and (\ref{7.2.15}): \begin{equation} \rank (d\pi_x)(\bar{z})=d-1. \label{eq-7.2.14} \end{equation} Then (after rotating coordinates) we can use $\tilde{S}(x',p_d)$ with $x'=(x_1,\ldots,x_{d-1})$ and \begin{equation} \tilde{S}_{p_dp_d}(\bar{z})= 0, \qquad \tilde{S}_{p_dp_dp_d}(\bar{z})\ne 0 \label{eq-7.2.15} \end{equation} We are interested in asymptotics of \begin{equation} u_h(x,t)= (2\pi h)^{-\frac{1}{2}} \int e^{ih^{-1}(p_dx_d -\tilde{S}(x',p_d,t)}\tilde{A}(x',p_d, t)\,dp_d \label{eq-7.2.16} \end{equation} near such point. One can prove that under assumptions (\ref{eq-7.2.14})--(\ref{eq-6.2.15}) \begin{multline} u_h(x,t)= (2\pi )^{-\frac{1}{2}}h^{-\frac{1}{6}} e^{ih^{-1}\beta (x,t)} \int e^{i(\frac{1}{3}q^3-h^{-\frac{2}{3}}\alpha (x,t)q)} B(qh^{\frac{1}{3}}, x,t)\,dq\sim\\ e^{ih^{-1}\beta (x,t)} \operatorname{Ai} (-h^{-\frac{2}{3}}\alpha(x,t)) c(x,t) \label{eq-7.2.17} \end{multline} with $\alpha_x \ne 0$. Here $\operatorname{Ai}$ is Airy function and as $|\alpha (x,t)|\gg h^{\frac{2}{3}}$, $\alpha <0$ we have $u_h \sim 0$ and $|\alpha (x,t)|\gg h^{\frac{2}{3}}$, $\alpha >0$ we have a corresponding asymptotic formula via exponents $e^{ih^{-1}S^\pm (x,t)}b_\pm (x,t,h)$ with $S^\pm (x,t)=\beta (x,t)\pm \frac{2}{3}\alpha(x,t)^{\frac{3}{2}}$.

$\Leftarrow$  $\Uparrow$  $\Rightarrow$