### A splitting formula for the spectral flow of the odd signature operator on 3-manifolds coupled to a path of $SU\left(2\right)$ connections.

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In this paper we consider a smooth and bounded domain $\Omega \subset {\mathbb{R}}^{d}$ of dimension $d\ge 2$ with boundary and we construct sequences of solutions to the wave equation with Dirichlet boundary condition which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. Moreover, the result holds for microlocally strictly convex domains...

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by...

We study the high-energy eigenfunctions of the Laplacian on a compact Riemannian manifold with Anosov geodesic flow. The localization of a semiclassical measure associated with a sequence of eigenfunctions is characterized by the Kolmogorov-Sinai entropy of this measure. We show that this entropy is necessarily bounded from below by a constant which, in the case of constant negative curvature, equals half the maximal entropy. In this sense, high-energy eigenfunctions are at least half-delocalized....

Given a one-parameter family $\{{g}_{\lambda}:\lambda \in [a,b]\}$ of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials $\{{V}_{\lambda}:\lambda \in [a,b]\}$ and a family $\{{\sigma}_{\lambda}:\lambda \in [a,b]\}$ of trajectories connecting two points of the mechanical system defined by $({g}_{\lambda},{V}_{\lambda})$, we show that there are trajectories bifurcating from the trivial branch ${\sigma}_{\lambda}$ if the generalized Morse indices $\mu \left({\sigma}_{a}\right)$ and $\mu \left({\sigma}_{b}\right)$ are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate...