Semiclassical Asymptotics 4


### Initial and Initial-Boundary Value Problems

#### IVP

We are interested in solving problem \begin{align} &P(x,-ih \nabla , h)u=0, \label{eq-5.4.1}\\ &(-ih\partial_t)^j u|_{t=0}= f_j(x')&& j=0,\ldots, m-1 \label{eq-5.4.2} \end{align} where $x=(x_0,x_1,\ldots, x_d)$, $t=x_0$, $x'=(x_1,\ldots, x_d)$, $m$ is maximal degree of $-ih\partial_t$ in $P$ (and also in $P_0$); other derivatives can have larger degrees.

For Schrödinger equation (5.1.3) $m=1$, for wave equation (5.1.1).

We assume that initial functions $f_k(x')$ are of the form $$f_j(x')\sim e^{ih^{-1}S_0(x')}\sum_{n\ge 0} F_{j,n} (x') h^n \label{eq-5.4.3}$$ (the same phase but different amplitudes). We are looking for solution in the form $$u(x)\sim \sum_{1\le k\le m} e^{ih^{-1}S_k(x)}\sum_{n\ge 0}A_{n,k} (x) h^n. \label{eq-5.4.4}$$ We know that $S_k(x)$ must satisfy (5.2.1) $$P_0(x,\nabla S_k(x))=0. \label{eq-5.4.5}$$

Assumption 1. All roots $p_0=-H_k (x,p')$, $k=1,\ldots, m$ of the polynomial $P_0(x,p_0, p')$ are real and distinct as $p'=\nabla' S_0(x')$.

Obviously this condition is fulfilled for Schrödinger equation, and it is fulfilled for wave equation as $\nabla' S_0(x')\ne 0$.

Under Assumption 1 we can find (locally) functions $S_k(x)$ satisfying (5.2.3) $$\partial_tS_k + H_k(x,\nabla 'S_k)=0\qquad k=1,\ldots,m. \label{eq-5.4.6}$$ Then $A_{k,n}$ must satisfy transport equations (5.3.1) (as $n=0$) and similar $$\bigl(\partial_t+\sum_{1\le j\le d} H_k^{(j)} (x, \nabla S(x))\partial_j +i Q_k(x)\bigr)A_{k,n} (x)= \sum_{l\le n-1} \mathcal{L}_{k,n-l} A_{k,l}. \label{eq-5.4.7}$$ Here $A_{k,n}$ are defined recurrently. To define $A_{k,n}$ we need their values as $t=0$ (and then we use transport equations).

Consider first $n=0$. Then (\ref{eq-5.4.2})--(\ref{eq-5.4.3}) are fulfilled modulo $O(h)$ if $$\sum_{1\le k\le m} (\partial_t S_k)^j A_{k,0}\bigl|_{t=0}=F_{j,0}\qquad j=0,\ldots, m-1. \label{eq-5.4.8}$$ This is a linear $m\times m$ system with Vandermonde matrix and the determinant $$\prod _{1\le j< k\le m} \bigl(-H_k (x',\nabla'S_0)+ H_k (x',\nabla'S_0)\bigr)\ne 0 \label{eq-5.4.9}$$ where we used (\ref{eq-5.4.6}) and Assumption 1.

Assume that we defined $A_{k,l}$ for all $l=0,\ldots, n-1$ and all $k=1,\ldots, m$ so that initial conditions are fulfilled modulo $O(h^n)$. Then for $A_{k,n}$ we get $$\sum_{1\le k\le m} (\partial_t S_k)^j A_{k,n}\bigl|_{t=0}=G_{j,n}\qquad j=0,\ldots, m-1 \label{eq-5.4.10}$$ where $G_{j,n}$ are $F_{j,n}$ and what came from $A_{k,l}$ with $l=0,\ldots, n-1$, $k=1,\ldots, m$.

Again determinant is not $0$.

#### IBVP. Reflection

Consider specifically wave equation (5.1.1) in domain $X$: $$u_{tt}-\nabla \cdot (c^2(x)\nabla u)=0 \label{eq-5.4.11}$$ There is a solution $$u\sim e^{ih^{-1}\phi (x)}\sum_{n\ge 0} A_k (x)h^n \label{eq-5.4.12}$$ where we prefer to denote eikonal by $\phi$ (etc). But this solution does not satisfy Dirichlet boundary condition $$u|_Y=0 \label{eq-5.4.13}$$ or Neumann or Robin boundary condition $$(\boldsymbol{\nu}\cdot \nabla +\kappa) u|_Y=0 \label{eq-5.4.14}$$ where $Y=\partial X$ and $\boldsymbol{\nu}$ is a normal to $Y$ directed into $X$.

Assumption 2. It is incoming wave which means that as $t$ increases trajectories come from $X$ to $Y$. This is equivalent to $$\frac{\partial \phi}{\partial\boldsymbol{\nu}} : \frac{\partial \phi}{\partial t} \Bigl|_Y >0. \label{eq-5.4.15}$$ where $\frac{\partial \phi}{\partial\boldsymbol{\nu}} =\boldsymbol{\nu}\cdot \nabla$.

To fulfill boundary condition we add a reflected wave $$u\sim e^{ih^{-1}\phi (x)}\sum_{n\ge 0} A_k (x)h^n + e^{ih^{-1}\psi (x)}\sum_{n\ge 0} B_k (x)h^n. \label{eq-5.4.16}$$ Both $\phi$ and $\psi$ must satisfy eikonal equation $$\phi_t^2 = c^2|\nabla \phi|^2,\qquad \psi_t^2 = c^2|\nabla \psi|^2; \label{eq-5.4.17}$$ they must coincide of $Y$: $$\psi=\phi \qquad \text{on }\ Y \label{eq-5.4.18}$$ and differ. Then $$\frac{\partial \psi}{\partial\boldsymbol{\nu}}= -\frac{\partial \phi}{\partial\boldsymbol{\nu}} \text{on }\ Y \label{eq-5.4.19}$$ Then the added term is an outgoing wave $$\frac{\partial \psi}{\partial\boldsymbol{\nu}} : \frac{\partial \psi}{\partial t} \Bigl|_Y < 0. \label{eq-5.4.20}$$ From this and equations for rays follows the well known rule: reflection angle equals incidence angle.

Remark 1. One can understand this from toy-model $c=\const$ and $X=\{x_1>0\}$, $\phi = ct - k x_1- l x_2$; then $\psi = ct + k x_1- l x_2$ ($k^2+ l^2=1$).

Picture will be here later

Now Dirichlet or Robin boundary condition imply that $$B_0= \mp A_0\quad\text{on }\ Y \label{eq-5.4.21}$$ and similar conditions for $B_n$. Since $B_n$ must satisfy transport equations we can find them locally.

#### IBVP. Reflection and Refraction

Consider now two wave equations (5.1.1) in domains $X_1$ and $X_2$: $$u_{j,tt}-\nabla \cdot (c_j^2(x)\nabla u_j)=0 \quad\text{in }\ X_j. \label{eq-5.4.22}$$ These domains have a common boundary $Y$ where two boundary conditions connecting $u_1$ and $u_2$ must be satisfied. We consider incoming wave (\ref{eq-5.4.12}) in $X_1$ satisfying (\ref{eq-5.4.15}) where $\boldsymbol{\nu}$ is a normal to $Y$ directed into $X_1$.

To satisfy two boundary conditions on $Y$ we need not only consider reflected wave (so solution is given by (\ref{eq-5.4.16}) in $X_1$ but also a refracted wave $$u\sim e^{ih^{-1}\chi (x)}\sum_{n\ge 0} C_k (x)h^n\quad\text{in }\ X_2. \label{eq-5.4.23}$$ Here $\chi$ must satisfy eikonal equation $\chi_t^2=c_2^2|\nabla \chi|^2$ and correspond to outgoing wave. From this and equations for rays follows the well known Snell law: $$\frac{\sin(\alpha_1)}{c_1}=\frac{\sin(\alpha_2)}{c_2} \label{eq-5.4.24}$$ where $\alpha_1$ is an angle between incidental angle and normal and $\alpha_2$ is an angle between reflection angle and normal.

Remark 2. One can understand this from toy-model $c_j=\const$ and $X_1=\{x_1>0\}$, $X_2=\{x_1< 0\}$, $\phi = c_1t - k x_1- l x_2$; then $\psi = c_t + k x_1- l x_2$ ($k^2+ l^2=1$) but $\chi = c_1 t +m x_1 -l x_2$; then $k^2+l^2=1$ but $m^2 + l^2= c_1^2/c_2^2$. Clearly $l=\sin (\alpha_1)=\sin (\alpha_2) c_2/c_1$.

Picture will be here later

Remark 3. It may happen that we cannot find real $\alpha_2$ as $$\frac{c_2\sin(\alpha_1)}{c_1} >1. \label{eq-5.4.25}$$ Then we cannot find real valued $\chi$ but we can find complex-values $\chi$ with $\Im \chi > 0$. Then we have a wave which exponentially decays into $X_2$ penetrating there on the depth $\asymp h$. Geometrically this wave is not observable (it is where incoming wave hits $Y$) but analytically it is still here.

This is called a complete internal reflection.

Remark 4. In elasticity theory there are Rayleigh waves propagating along the free surface and Lamb waves propagating along the surface separating two media.