I've been working on Bourgain's paper 'Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces' for my Master's thesis and everything was going great until I reached the last section where he begins with applications on the Laplacian; and introduces $e^{it\Delta} $, the periodic Schrödinger group ... 


I realize this might not be the best place to ask this; and I tried searching Bourgain's earlier papers for a definition to no avail, so would someone please tell me what $e^{it\Delta} $ is?



The Periodic Schrödinger Group 
$e^{it\Delta}$ is the Fourier multiplier $e^{-4it\pi^2\vert D\vert^2}$, i.e. the operator defined by
$$
(e^{it\Delta} u)(x)=\int\_{\mathbb R ^d} e^{2i\pi x \xi}e^{-4it\pi^2\vert \xi\vert^2}\hat u(\xi) d\xi.
$$
It is also the convolution with $E(t)$, say for $t>0$,
$$
E(t)(x)=e^{-i(d-2)\pi/4}(4\pi t)^{-d/2}e^{i\vert x\vert^2 /(4t)},\quad
(e^{it\Delta} u)(x)=(E(t)\ast u)(x).
$$
The Fourier multiplier definition is the simplest.


