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

Problems to Chapter 5

1. Problem 1
2. Problem 2
3. Problem 3
4. Problem 4
5. Problem 5
6. Problem 6
7. Problem 7

Problem 1.

For 1D-problem \begin{align} &u_{tt}-u_{xx}=0,\\ &u|_{t=0}=e^{ikx},\\ &u_t|_{t=0}=-ike^{ikx} \end{align}

1. write eikonal equation for phase $\phi$ and solve it.
2. write transport equations and solve them.

Problem 2.

For 1D-problem \begin{align} &u_{tt}-x^2 u_{xx}=0,\\ &u|_{t=0}=e^{ikx},\\ &u_t|_{t=0}=-ikxe^{ikx} \end{align}

1. write eikonal equation for phase $\phi$ and solve it.
2. write transport equation for $A_0$ and solve it.

Problem 3.

For 2D-problem \begin{align} &u_{tt}-\Delta u=0,\\ &u|_{t=0}=e^{ik\sqrt{x^2+y^2}},\\ &u_t|_{t=0}=ike^{ik\sqrt{x^2+y^2}} \end{align}

1. write eikonal equation for phase $\phi$ and solve it.
2. write transport equation for $A_0$ and solve it.

Problem 4.

For 3D-problem \begin{align} &u_{tt}-\Delta u=0,\\ &u|_{t=0}=e^{ik\sqrt{x^2+y^2+z^2}},\\ &u_t|_{t=0}=-ike^{ik\sqrt{x^2+y^2+z^2}} \end{align}

1. write eikonal equation (for phase $\phi$ and solve it.
2. write transport equation for $A_0$ and solve it.

Problem 5.

For 1D-equation \begin{align} - h^2 u_{xx} +V(x) u=Eu , \end{align}

1. write equation for phase $\phi$;
2. write dynamical system.
3. Consider $V(x)=x^2$.
4. Consider $V(x)=|x|$.

Problem 6.

For 3D-equation \begin{align} - h^2 \Delta u +V(\boldsymbol{x}) u=Eu , \end{align}

1. write equation for phase $\phi$;
2. write dynamical system.
3. Consider $V(\boldsymbol{x})=|\boldsymbol{x}|^2$.
4. Consider $V(\boldsymbol{x})=|\boldsymbol{x}|$.

Problem 7.

For 3D-equation \begin{align} \bigl(-ih \nabla - \boldsymbol{A}\bigr) ^2 u +V(x) u=Eu , \end{align}

1. write equation for phase $\phi$;
2. write dynamical system.
3. Consider $V(\boldsymbol{x})=|\boldsymbol{x}|^2$, $\boldsymbol{A}= x_1\boldsymbol{i}$.
4. Consider $V(\boldsymbol{x})=0$, $\boldsymbol{A}= x_2 \boldsymbol{i}$.
5. Consider $V(\boldsymbol{x})=|\boldsymbol{x}|^2$, $\boldsymbol{A}= x_2\boldsymbol{i}$.

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