Semiclassical Asymptotics 1

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Chapter 5. Semiclassical and High Frequency Asymptotics: Local Theory


  1. Introduction
  2. Action on exponent


We are interested in high-frequency asymptotics of wave equation (and similar) \begin{equation} u_{tt}-\nabla \cdot (c^2(x)\nabla u)=0 \label{eq-5.1.1} \end{equation} and semiclassical asymptotics to Schrödinger equation (and similar) \begin{equation} -i\hbar u_t -\frac{\hbar^2}{2m} \Delta u + V(x)u=0. \label{eq-5.1.2} \end{equation} By high-frequency solutions we mean solutions to (\ref{eq-5.1.1}) $u=e^{ik \phi (x,t)}A(x,t,k^{1})$ with $k\gg 1$ and by semiclassical asymptotics we mean solutions to (\ref{eq-5.1.2}) with $\hbar \ll 1$, $m\asymp 1$. In fact if $m \ll 1$ we divide to (\ref{eq-5.1.2}) by $m$ we get \begin{equation} -ih u_t -\frac{h^2}{2} \Delta u + \frac{1}{m}V(x)u=0. \label{eq-5.1.3} \end{equation} with $h= \hbar/2m$ and usually $ \frac{1}{m}V(x)\asymp 1$ and we interested in asymptotics when $h\ll 1$. So, we consider (\ref{eq-5.1.3}). Further, (\ref{eq-5.1.1}) after division by $k^2$ and introducing $h=k^{-1}$ becomes of the similar form \begin{equation*} h^2u_{tt}-h\nabla \cdot (c^2(x)h\nabla u)=0. \end{equation*} We are looking for solutions to equation \begin{equation} P(x, ih\nabla , h)u=0 \label{eq-5.1.4} \end{equation} where now we include $t=x_0$ in $x$ and $u=e^{ih^{-1}S(x)} A(x,h^{-1})$.

Here \begin{equation} P:= \sum_{\alpha:|\alpha|\le m, l} a_{\alpha,l} (-ih\nabla )^\alpha h^l \label{eq-5.1.5} \end{equation} where $\alpha=(\alpha_0,\ldots ,\alpha_d)$ is multinidex, $|\alpha|=\alpha_0+\ldots+\alpha_d$, $\alpha!=\alpha_0!\cdots \alpha_d!$ and $p^\alpha= p_0^{\alpha_0}\cdots p_d^{\alpha_d}$.

Remark 1 in Quantum Mechanics $-ih\nabla$ is a momentum operator.

Action on exponent

Theorem 1 \begin{multline} e^{-ih^{-1}S(x)} P \bigl(e^{ih^{-1}S(x)} A(x)\bigr)= P_0 (x,\nabla S(x)) A +\\ h \bigl(-i\sum_j P_0^{(j)} (x, \nabla S(x))\partial_j +Q(x)\bigr)A (x)+\ldots \label{eq-5.1.6} \end{multline} where dots denote terms with higher powers of $h$ and \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-5.1.7} \end{equation} and we use notations \begin{equation} P_l(x,p):= \sum_{\alpha:|\alpha|\le m} a_{\alpha,l} (x)p^\alpha . \label{eq-5.1.8} \end{equation} $P^{(j)}(x,p)=\partial_{p_j} P(x,p)$, $P_{(k)}(x,p)=\partial_{x_k} P(x,p)$ etc.

Proof. Think about it!

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