TITLE: A Symplectic Approach to Hardy's Uncertainty Principle.

ABSTRACT. A folk metatheorem is that a function $f$ and its Fourier transform cannot be simultaneously sharply localized. One way to express this kind of trade-off between $\psi$ and $\widehat{\psi}$ was proven in 1933 by G. H. Hardy [2] who showed, using the Phragmén-Lindelöf principle, that if $\psi\in L^{2}(\mathbb{R})$ and its Fourier transform satisfy, for $\vert x\vert,\vert p\vert\rightarrow\infty$, estimates $f(x)=\mathcal{O}% (e^{-ax^{2}/2})$, $\widehat{\psi}(p)=\mathcal{O}(e^{-bp^{2}/2})$ with $a,b>0$, then the following holds:

(1)

If $ab>1$ then $f=0$;

(2)

If $ab=1$ we have $f(x)=Ce^{-ax^{2}/2}$ for some complex constant $C$;

(3)

If $ab<1$ we have $f(x)=Q(x)e^{-ax^{2}/2}%$ where $Q$ is a polynomial function.

In this talk I will use Hardy's uncertainty principle to analyze the condition for a Gaussian in $(x,p)$ space to be the Wigner transform of a positive trace-class operator (i.e. a mixed state in the language of quantum mechanics). I will formulate the result in terms of the topological notion of symplectic capacity of the associated Wigner ellipsoid. (Joint work with Franz Luef (Vienna)).