Exercise 33, § 5

Let $n\in \mathbf{N}$. We denote by Δ the Laplacian on $L^2(\mathbf{R}^n)$.

a) For every $t\in \mathbf{R}^*$, the endomorphism $e^{it\Delta}$ of $L^2(\mathbf{R}^n)$ is given by

if $f\in \mathscr{L}^1(\mathbf{R}^n)\cap \mathscr{L}^2(\mathbf{R}^n)$. (Use the Fourier transform.)

b) Let C be the unitary mapping on $L^2(\mathbf{R}^n)$ deduced by passing to the quotient from the mapping $f\mapsto C(f)$ defined for $f\in \mathscr{L}^2(\mathbf{R}^n)$ by

for $x\in \mathbf{R}^n$.

One has $e^{it\Delta}\rightarrow C$ when $t\rightarrow +\infty$ in the space $\mathscr{L}(L^2(\mathbf{R}^n))$ endowed with the topology of simple convergence.

c) Suppose that $1\leqslant n\leqslant 3$. Let $g\in \mathscr{L}^{\infty}(\mathbf{R}^n) +\mathscr{L}^2(\mathbf{R}^n)$ be real-valued. Let us denote by $u$ the partial operator $u= \Delta +m_g$; it is self-adjoint on $L^2(\mathbf{R}^n)$ (exercise 14). For $f\in \mathscr{L}^1(\mathbf{R}^n)\cap \mathscr{L}^2(\mathbf{R}^n)$ and $t\in \mathbf{R}^*$, one has

$e^{itu}f(x) =$ lim$_{k\rightarrow+\infty}\frac{1}{(2it/k)^{kn/2}}\int_{\mathbf{R}^{nk}}f(y_k)$ exp$(\frac{it}{k}\sum_{i=1}^k(\frac{|y_i-y_{i-1}|^2}{4(t/k)^2}-g(y_i))dy$

where one writes $y= (y_1, . . . , y_k)$ with $y_i\in \mathbf{R}^n$.

d) Interpretation of the preceding formula.