$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\bC}{\mathbb{C}}$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\const}{\operatorname{const}}$ ###[Transport equations](id:sect-5.3) > 1. [First transport equation](#sect-5.3.1) > 2. [Density property](#sect-5.3.2) > 3. [Next transport equations](#sect-5.3.3) #### [First transport equation](id:sect-5.3.1) As $S(x,t)$ satisfies equation equation [(5.2.1)](./L5.2.html#mjx-eqn-eq-5.2.1) \begin{equation\*} P\_0(x,\nabla S(x))=0 \end{equation\*} (we again consider $x\_0=t$ as part of $x$) consider the next term in the right-hand expression [(5.1.6)](./L5.1.html#mjx-eqn-eq-5.1.6); it has factor $h$ and we equalize it to $0$ ignoring so far all other terms: \begin{equation} \bigl(-i\sum\_j P\_0^{(j)} (x, \nabla S(x))\partial\_j +Q(x)\bigr)A (x)=0 \label{eq-5.3.1} \end{equation} where \begin{equation} Q(x)=-\frac{i}{2}\sum \_{j,k} P\_0^{(jk)}(x,\nabla S(x)) S\_{x\_jx\_k} (x) +P\_1(x,\nabla S(x)) \label{eq-5.3.2} \end{equation} Equation (\ref{eq-5.3.1}) to *amplitude* $A$ is a linear ODE along trajectory [(5.2.5)](./L5.1.html#mjx-eqn-eq-5.1.6). Indeed, \begin{equation\*} \sum\_j P\_0^{(j)} (x, \nabla S(x))\partial\_j = \frac{d\ }{d\tau}; \end{equation\*} so (\ref{eq-5.3.1}) becomes \begin{equation} \bigl(\frac{d\ }{d\tau} +iQ(x)\bigr)A (x)=0 \label{eq-5.3.3} \end{equation} Let us look at $Q(x)$. Observe that according to Liouville theorem along trajectories \begin{multline} \frac{d\ }{d\tau} \ln J(x,t) = \sum\_j \partial\_{x\_j} \bigl(P\_0^{(j)}(x,\nabla S(x))\bigr)=\\\\ \sum\_j \partial\_{x\_j} P\_{0(j)}^{(j)}(x,\nabla S(x)) + \sum\_{j,k} P\_{0}^{(jk)}(x,\nabla S(x))S\_{x\_jx\_k} \label{eq-5.3.4} \end{multline} where $J$ is a volume element. Recall that $P\_{0(j)}$ is calculated as if $\nabla S$ does not depend on $x$. Then (\ref{eq-5.3.3}) becomes \begin{equation} \bigl(\frac{d\ }{d\tau} +i P^s(x,\nabla S)\bigr)A (x)J^{1/2}=0 \label{eq-5.3.5} \end{equation} with \begin{equation} P^s := P\_1 -\frac{i}{2} \sum\_j P\_{0(j)}^{(j)}. \label{eq-5.3.6} \end{equation} ####[Density property](id:sect-5.3.3) In particular \begin{equation} \bigl(\frac{d\ }{d\tau} +2\Re i P^s(x,\nabla S)\bigr)|A (x)|^2J=0. \label{eq-5.3.7} \end{equation} **[Remark 1](id:rem-5.3.1).** In fact $P^s$ rather than $P\_1$ is a "correct" expression. All symbols $P\_l$ with $l\ge 1$ depend on he *method of quantization*. We used quantization in which $p=-ih\nabla$ acts first and then $q$ (aka $x$). But this violates lot of things. Correct quantization was invented by Hermann Weyl and is called *Weyl quantization* (or *symmetric quantization*) and in this quantization $P^s$ is the next symbol! Usually in *Microlocal analysis* $P\_0$ is called *principal symbol* and $P^s$ is called *subprincipal symbol*. For Hermitian and self-adjoint operators all Weyl symbols are real (and v.v.) so in this case $\frac{d\ }{d\tau} |A|^2J=0$ and $|A|^2$ is a *density*. ####[Next transport equations](id:sect-5.3.3) In fact there are many terms in the right-hand expression [(5.1.6)](./L5.1.html#mjx-eqn-eq-5.1.6). Because of this we need amplitude $A(x)$ in the form \begin{equation} A(x)\sim \sum\_{n\ge 0} A\_0(x)h^n \label{eq-5.3.8} \end{equation} and $A\_0$ rather than $A$ satisfies all above equations while $A\_n$ with $n\ge 1$ takes care of the term with $h^{n+1}$: \begin{equation} \bigl(\frac{d\ }{d\tau} +iQ(x)\bigr)A \_n(x)=\sum\_{l\le n-1} \mathcal{L}\_{n-l} A\_l \label{eq-5.3.9} \end{equation} where $\mathcal{L}\_{n-l}$ are partial differential operators of order $n-l+1$. ________ [$\Leftarrow$](./L5.2.html) [$\Uparrow$](../contents.html) [$\Rightarrow$](./L5.4.html)