This paper is a continuation of Part I of the same title which has appeared at the last issue of this journal.

On construit, sur une variété riemannienne $X$ de dimension $2$ ou $3$, les extensions autoadjointes ${\Delta }_{\alpha ,x_0}(\alpha \in \mathbf{R}/\pi \mathbf{Z})$ de la restriction du laplacien aux fonctions nulles au voisinage d’un point $x_0$ de $X$. On calcule explicitement les valeurs propres de ${\Delta }_{\alpha ,x_0}$.

Dans cet article, nous étudions une famille d’opérateurs auto-adjoints ${\Delta }_a$ dérivés du laplacien sur une surface de Riemann d’aire finie et ayant au voisinage de l’infini la structure d’un cylindre $[b,+\infty [\times \mathbf{R}/\mathbf{Z}$ muni d’une métrique à courbure constante $-1$. Après avoir étudié la théorie spectrale de tels opérateurs, nous donnons, comme application, un théorème prévoyant l’absence générique de valeurs propres immergées dans le spectre continu du laplacien de ces surfaces. Nous montrons enfin comment ceci permet de...

We present here a simplified version of recent results obtained with B. Helffer and M. Klein. They are concerned with the exponentally small eigenvalues of the Witten Laplacian on $0$-forms. We show how the Witten complex structure is better taken into account by working with singular values. This provides a convenient framework to derive accurate approximations of the first eigenvalues of ${\Delta }_{f,h}^{\left(0\right)}$ and solves efficiently the question of weakly resonant wells.

We construct pairs of compact Kähler-Einstein manifolds $(M_i,g_i,{\omega }_i)(i=1,2)$ of complex dimension $n$ with the following properties: The canonical line bundle $L_i={\bigwedge }^n{T}^{*}{M}_i$ has Chern class $[{\omega }_i/2\pi ]$, and for each positive integer $k$ the tensor powers $L_1^{\otimes k}$ and $L_2^{\otimes k}$ are isospectral for the bundle Laplacian associated with the canonical connection, while $M_1$ and $M_2$ – and hence $T^{*}{M}_1$ and $T^{*}{M}_2$ – are not homeomorphic. In the context of geometric quantization, we interpret these examples as magnetic fields which are quantum equivalent but not classically equivalent....

We study the discrete groups $\Lambda$ whose duals embed into a given compact quantum group, $\widehat{\Lambda }\subset G$. In the matrix case $G\subset U_n^{+}$ the embedding condition is equivalent to having a quotient map ${\Gamma }_U\to \Lambda$, where $F=\{{\Gamma }_U\mid U\in U_n\}$ is a certain family of groups associated to $G$. We develop here a number of techniques for computing $F$, partly inspired from Bichon’s classification of group dual subgroups $\widehat{\Lambda }\subset S_n^{+}$. These results are motivated by Goswami’s notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian...

In these notes, we will describe recent work on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr-de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non-elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally...

The conformal infinity of a quaternionic-Kähler metric on a $4n$-manifold with boundary is a codimension $3$ distribution on the boundary called quaternionic contact. In dimensions $4n-1$ greater than $7$, a quaternionic contact structure is always the conformal infinity of a quaternionic-Kähler metric. On the contrary, in dimension $7$, we prove a criterion for quaternionic contact structures to be the conformal infinity of a quaternionic-Kähler metric. This allows us to find the quaternionic-contact structures...