HW Exercises: If you hand them in, I will grade them

The first two exercises are from Grigis-Sjostrand. Although they are at the end of the book, you don't need anything from earlier chapters to do them.

1.  Do exercise 12.1 of  Grigis-Sjostrand: WKB in dimenson one. Why is there no quantization condition?

2. Do exercise 12.2, which I copy out here if you don't have the book.

HW on Stationary Phase:

1. The Gamma function (factorial function) is defined by $\Gamma(x + 1) = \int_0^{\infty} e^{-t} t^x dt. $ Stirling's formula gives the asymptotics of $\Gamma(x  + 1)$ as  $x \to \infty$.

Determine them using the stationary phase method (really, the steepest descent method since there is no $i$ in the phase). Hint: change variables to that the integrand has the form $e^{x F(t)}$ for some function

$F(t)$. What is $F$? Justify all the steps.

2. Consider the Schrodinger operator $$\hat{H} = - \frac{h^2}{2} \frac{d^2}{dx^2} + V $$
on $L^2(\R)$. Apply  $\hat{H} - E $ to the oscillatory integral $$\psi_h(x) = h^{-1/2}  \int e^{\frac{i}{h} (\langle x, \xi \rangle + T(\xi))}
a_h(x, \xi) d \xi. $$

Use the stationary phase method to show that $(\hat{H} - E)  \psi_h = O(h)$  if and only if $x = T ' (\xi)$ parameterizes the energy curve $\xi^2 + V(x) = E. $

3. The $J_0(x)$ Bessel function is the oscillatory integral $J_0(x) = \frac{1}{\pi}|  \int_0^{\pi} \cos (x \sin \phi) d\phi $. Determine the asymptotics of $J_0(x)$ as $x \to \infty$ by apply the statioary phase method.

4. Consider the upside down classical harmonic oscillator $H(x, \xi) = \frac{1}{2}(\xi^2 - x^2)$ and its quantization $\hat{H} = - \frac{h}{2} \frac{d^2}{dx^2} - x^2/2$.

Find its propagator $U_h(t, x, y) = A_h(t, x, y) e^{i/h S(t, x, y)}$. Solve explicitly the Hamilton Jacobi equation for $S$: $\frac{\partial S}{\partial t} + H(x, \partial S/\partial x) = 0$.

Write out and solve the transport equation for $A$.

5. Solve the Hamilton Jacobi equation $|\nabla S(x) | = 1$ in $\R^2$, $S = 0$ on the unit circle  in $\R^2$. What is its geometric interpretation.

6. Do Problem 8 in Chapter 3.5 of Evans PDE: Let $E \subset \R^n$ be closed. Solve the initial value problem for the HJ equation $u_t + |Du(x)|^2 = 0$

with initial condition $u(0, x) = 0 (x \in E), + \infty (x \notin E). $ Show that $u(x, t) = (4t)^{-1} d(x, E)^2. $

6. 4/1/09:  Construct a solution of the wave equation of the form $\int_0^{\infty} e^{i \theta (t - r(x, y)} A(x, y, \theta) d\theta$. I.e. use the Hadamard method

but with phase $t - r$ instead of $r^2 - t^2$. What are the new transport equations. Again write $A = \theta^{\frac{d-1}{2}} \sum U_j(x, y) \theta^{-j}. $