Given a complete Riemannian manifold $M$ and a Schrödinger operator $-\Delta+m$ acting on $L^p(M)$, we study two related problems on the spectrum of $-\Delta+m$. The first one concerns the positivity of the $L^2$-spectral lower bound $s(-\Delta+m)$. We prove that if $M$ satisfies $L^2$-Poincaré inequalities and a local doubling property, then $s(-\Delta+m)>0$, provided that $m$ satisfies the mean condition

$\inf\substack {p\in M}\frac {1}{|B(p, r)|}\int \sb{B(p,r )}m(x)dx>0$

for some $r>0$. We also show that this condition is necessary under some additional geometrical assumptions on $M$.

The second problem concerns the existence of an $L^p$-principal eigenvalue, that is, a constant $\lambda\geq 0$ such that the eigenvalue problem $\Delta u=\lambda mu$ and equation above] has a positive solution $u\in L^p(M)$. We give conditions in terms of the growth of the potential $m$ and the geometry of the manifold $M$ which imply the existence of $L^p$-principal eigenvalues.

Finally, we show other results in the cases of recurrent and compact manifolds.

In this paper, we consider real hypersurfaces $M$ in $\mathbb {C}^3$ (or, more generally, $5$-dimensional CR (Cauchy-Riemann) manifolds of hypersurface type) at uniformly Levi degenerate points, that is, Levi degenerate points such that the rank of the Levi form is constant in a neighborhood. We also require that the hypersurface satisfy a certain second-order nondegeneracy condition (called $2$-nondegeneracy) at the point. One of our main results is the construction, near any point $p_0\in M$ satisfying the above conditions, of a principal bundle $P\to M$ and a $\mathbb {R}^{\dim P}$-valued 1-form $\underline {\omega}$, uniquely determined by the CR structure on $M$, which defines an absolute parallelism on $P$. If $M$ is real-analytic, then covariant derivatives of $\underline {\omega}$ yield a complete set of local biholomorphic invariants for $M$. This solves the biholomorphic equivalence problem for uniformly Levi degenerate hypersurfaces in $\mathbb {C}^3$ at $2$-nondegenerate points.

Nous étudions le corps $K_p$ engendré par les points d'ordre premier $p$ d'une courbe elliptique sur un corps de nombres. Avec l'aide de Kowalski et Michel, nous démontrons que pour presque tout nombre premier $p$, toute courbe elliptique sur le corps cyclotomique $\mathbf {Q}(\mu_p)$ possédant un sous-groupe cyclique $\mathbf {Q}(\mu_p)$-rationnel d'ordre $p$ a potentiellement bonne réduction en caractéristique $p$. Nos méthodes s'appliquent aussi pour étudier, nombre premier par nombre premier, les courbes elliptiques n'ayant pas potentiellement bonne réduction en $p$. En particulier, nous démontrons qu'on a $K_p\neq \mathbf {Q}(\mu_p)$ pour $5<p<1000$ et $p\equiv 1(\mod 4)$. Nous faisons un usage crucial des travaux de Kato sur la conjecture de Birch et Swinnerton-Dyer.

Let $\mathfrak {g}$ be a semisimple Lie algebra, and let $\mathfrak {h}$ be a reductive subalgebra of maximal rank in $\mathfrak {g}$. Given any irreducible representation of $\mathfrak {g}$, consider its tensor product with the spin representation associated to the orthogonal complement of $\mathfrak {h}$ in $\mathfrak {g}$. Recently, B. Gross, B. Kostant, P. Ramond, and S. Sternberg [2] proved a generalization of the Weyl character formula which decomposes the signed character of this product representation in terms of the characters of a set of irreducible representations of $\mathfrak {h}$, called a multiplet. Kostant [7] then constructed a formal $\mathfrak {h}$-equivariant Dirac operator on such product representations whose kernel is precisely the multiplet of $\mathfrak {h}$-representations corresponding to the given representation of $\mathfrak {g}$.

We reproduce these results in the Kac-Moody setting for the extended loop algebras $\tilde {L}\mathfrak {g}$ and $\tilde {L}\mathfrak {h}$. We prove a homogeneous generalization of the Weyl-Kac character formula, which now yields a multiplet of irreducible positive energy representations of $L\mathfrak {h}$ associated to any irreducible positive energy representation of $L\mathfrak {g}$. We construct an $L\mathfrak {h}$-equivariant operator, analogous to Kostant's Dirac operator, on the tensor product of a representation of $L\mathfrak {g}$ with the spin representation associated to the complement of $L\mathfrak {h}$ in $L\mathfrak {g}$. We then prove that the kernel of this operator gives the $L\mathfrak {h}$-multiplet corresponding to the original representation of $L\mathfrak {g}$.

We exhibit an intermittency phenomenon in quantum dynamics. More precisely, we derive new lower bounds for the moments of order $p$ associated to the state $\psi(t)=e^{-itH}\psi$ and averaged in time between zero and $T$. These lower bounds are expressed in terms of generalized fractal dimensions $D^\pm_{\mu_\psi}(1/(1+p/d))$ of the measure $\mu_\psi$ (where $d$ is the space dimension). This improves previous results obtained in terms of Hausdorff and Packing dimension.