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

#### Global construction

We considered equation $$-ih^{-1}u_t + H(x,hD_x,h) u=0. \label{eq-7.2.1}$$ We constructed solution $$u_h(x,t)= e^{ih^{-1}S(x,t)} A(x,t,h) \label{eq-7.2.2}$$ with $$A(x,t,h)\sim \sum_{n\ge 0} A_n (x,t)h^n \label{eq-7.2.3}$$ 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 $$(F_Ju)(x_I,p_J,t) = e^{-ih^{-1}\tilde{S}(x_I,p_J,t)} \tilde{A}(x_J,P_J,h) \label{eq-7.2.4}$$ where $\tilde{S}(p)$ is a partial Legendre transform of $S(x)$: $$\tilde{S}(p)= p_J\cdot x_J(x_I,p_J) - S(x_I, x_J) \label{eq-7.2.5}$$ 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 $$\tilde{A}_0(p)= \frac{1}{\sqrt{|S_{x_Jx_J}|}} e^{-\frac{i\pi}{4}\sgn(S_{x_Jx_J})} A(x_I,x_J). \label{eq-7.2.6}$$

Further, it solves (\ref{7.2.1}) in $(x_I,p_J)$-representation $$-ih^{-1}v_t + H(x_I,-hD_{p_J},hD_{x_I},p_J,h) v=0 \label{eq-7.2.7}$$ and therefore $\tilde{S}$ satisfies corresponding Hamilton-Jacobi equation $$-\tilde{S}_t + H_0(x_I,\tilde{S}_{p_J}, -\tilde{S}_{x_I},p_J)=0 \label{eq-7.2.8}$$ 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 $$u_h (x,t) = F_J^{-1}v = (2\pi h)^{-\frac{d-m}{2}} \ \int e^{ih^{-1} p_J\cdot x_J } v_h(x_I,p_J,t)\,dp \label{eq-7.2.9}$$ 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 $$a_0(z,t)= A_0(z,t)|\frac{dx}{dz}|^{\frac{1}{2}} \label{eq-7.2.10}$$ and $$\tilde{a_0}(z,t)= \tilde{A}_0(z,t)|\frac{dx}{dz}|^{\frac{1}{2}} \label{eq-7.2.11}$$ 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 $$e^{-\frac{i\pi}{4}\sgn(\frac{dp}{dx})} \label{eq-7.2.12}$$ 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