On donne une borne supérieur du nombre des valeurs propres négatives de l’opérateur de Schrödinger généralisé, cette borne est donnée en fonction d’un nombre fini de cube dyadiques minimaux.

In this survey, we present various forms of the uncertainty principle (Hardy, Heisenberg, Benedicks...). We further give a new interpretation of the uncertainty principles as a statement about the time-frequency localization of elements of an orthonormal basis, which improves previous unpublished results of H. Shapiro.Finally, we reformulate some uncertainty principles in terms of properties of the free heat and shrödinger equations.

We show that the differential inequality $\left|\Delta u\right|\le v\left|u\right|$ has the unique continuation property relative to the Sobolev space $H_{loc}^{2,1}\left(\Omega \right)$, $\Omega \subset R^n$, $n\le 3$, if $v$ satisfies the condition$$(K_n^{\mathrm{loc}})\underset{r\to 0}{lim}\underset{x\in K}{sup}{\int }_{|x-y|\<r}|x-y{|}^{2-n}v\left(y\right)dy=0$$for all compact $K\subset \Omega$, where if $n=2$, we replace $|x-y{|}^{2-n}$ by $-log|x-y|$. This resolves a conjecture of B. Simon on unique continuation for Schrödinger operators, $H=-\Delta +v$, in the case $n\le 3$. The proof uses Carleman’s approach together with the following pointwise inequality valid for all $N=0,1,2,...$ and any $u\in H_c^{2,1}({\mathbf{R}}^3-\left\{0\right\}),$$$...$$

Let us consider in a domain Ω of Rn solutions of the differential inequality|Δu(x)| ≤ V(x)|u(x)|, x ∈ Ω,where V is a non smooth, positive potential.We are interested in global unique continuation properties. That means that u must be identically zero on Ω if it vanishes on an open subset of Ω.

We show that phase space bounds on the eigenvalues of Schr¨odinger operators can be derived from universal bounds recently obtained by E. M. Harrell and the author via a monotonicity property with respect to coupling constants. In particular, we provide a new proof of sharp Lieb– Thirring inequalities.