Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 30E (Partial Differential Equations)

Use $\hat{f}(\xi )=\int e^{-ix\cdot \xi }f(x)dx$ with inverse factor $(2\pi {)}^{-n}$. The Fourier transform of the Schrödinger equation gives $i{\partial }_t\hat{\Psi }=|\xi {|}^2\hat{\Psi }/2$, hence $\hat{\Psi }(\xi ,t)=e^{-it|\xi {|}^2/2}\hat{f}(\xi )$. To invert the multiplier, first damp it by $e^{-\epsilon |\xi {|}^2}$. The Gaussian integral in each coordinate givesLet $\epsilon \downarrow 0$, with the branch continuous from positive real damping. The fundamental solution of a linear differential operator isFor Schwartz initial data this convolution and Fourier derivation are justified directly; the multiplier tends to one as $t\downarrow 0$ and has modulus one, giving the initial data and a unitary extension to $L^2$. The kernel itself is interpreted through the displayed Abel regularization.

Set $G(x,y,t)=\Psi (x+y/2,t)\overline{\Psi }(x-y/2,t)$. The Schrödinger equation and its conjugate givesince ${\nabla }_x={\nabla }_{+}+{\nabla }_{-}$ and ${\nabla }_y=({\nabla }_{+}-{\nabla }_{-})/2$. In the Wigner transform, integration by parts turns ${\nabla }_y$ into $i\xi$, yielding the free transport equationAlong its characteristics $x(t)=x_0+t\xi$, the value is constant. ThusFor rapidly decreasing smooth data every integration by parts is justified; weaker data follow distributionally.