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

Chapter 6 problems

• Problem 1
• Problem 3
• Problem 5
• Problem 7
• Problem 9
• Problem 11

Part I

Introduce sets

1. $\Theta: = \{(x,t)\colon \Phi_t(x,t)=0\}$;
2. $\Theta_{\text{sing}}: = \{(x,t)\colon \Phi_{tt} (x,t)=\Phi _{tx}(x,t) =0\}$;
3. Lagrangian manifold $\Lambda := \{(x,\Phi_x) \colon (x,t)\in \Theta \setminus \Theta_{\text{sing}}\}$;
4. $\pi_X \Lambda :=\{x\colon \exists t\ \Phi _t (x,t)=0\}$;
5. Causitic set $C_0:= \{x\colon \exists t\ \Phi _t (x,t)=\Phi _{tt} (x,t)=0 \}$.

Problem 1. For $\Phi (x,t)= t^3 -3tx$ find these sets.

Problem 2. For $\Phi (x,t)= 2t^3 -3t^2x$ find these sets.

Problem 3. For $\Phi (x,t)= t^4 -4t x$ find these sets.

Problem 4. For $\Phi (x,t)= t^4 - 2t^2 x$ find these sets.

Problem 5. For $\Phi (x,t)= 3t^4 - 4t^3 x$ find these sets.

Part II.

Consider equation \begin{gather} -h^2u_{xx}+ V(x) u= Eu \label{eq-6.P.1} \end{gather}

Problem 6. Calculate approximately eigenvalues of equation (\ref{eq-6.P.1}) with $V(x)=x^2$.

Problem 7. Calculate approximately eigenvalues of equation (\ref{eq-6.P.1}) with $V(x)=|x|$.

Problem 8. Calculate approximately eigenvalues $E > 0$ of equation (\ref{eq-6.P.1}) with $V(x)= \left\{\begin{aligned} & |x| && |x|<1,\\ & \infty && |x|>1. \end{aligned}\right.$

Problem 9. Calculate approximately eigenvalues $E > 0$ of equation (\ref{eq-6.P.1}) with $V(x)= \left\{\begin{aligned} & x^2 && |x|<1,\\ & \infty && |x|>1. \end{aligned}\right.$

Problem 10. Calculate approximately eigenvalues $E < 0$ of equation (\ref{eq-6.P.1}) with $V(x)= \left\{\begin{aligned} & -|x| && |x|<1,\\ & \infty && |x|>1. \end{aligned}\right.$

Problem 11. Calculate approximately eigenvalues $E < 0$ of equation (\ref{eq-6.P.1}) with $V(x)= \left\{\begin{aligned} & -x^2 && |x|<1,\\ & \infty && |x|>1. \end{aligned}\right.$

Problem 12. Calculate approximately eigenvalues $E < 0$ of equation (\ref{eq-6.P.1}) with $V(x)= \left\{\begin{aligned} & 0 && |x|<1,\\ & -1 &&1\le |x|\le 2,\\ & \infty && |x|>2. \end{aligned}\right.$

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