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Nonlinear Schrödinger equation on four-dimensional compact manifolds

Patrick Gérard, Vittoria Pierfelice (2010)

Bulletin de la Société Mathématique de France

We prove two new results about the Cauchy problem in the energy space for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadratic nonlinearities as well as a necessary and sufficient condition of global well-posedness for small energy...

Non-Sunada graphs

Robert Brooks (1999)

Annales de l'institut Fourier

We consider the question of whether there is a converse to the Sunada Theorem in the context of k -regular graphs. We give a weak converse to the Sunada Theorem, which gives a necessary and sufficient condition for two graphs to be isospectral in terms of a Sunada-like condition, and show by example that a strong converse does not hold.

Normal form of the wave group and inverse spectral theory

Steve Zelditch (1998)

Journées équations aux dérivées partielles

This talk will describe some results on the inverse spectral problem on a compact riemannian manifold (possibly with boundary) which are based on V. Guillemin's strategy of normal forms. It consists of three steps : first, put the wave group into a normal form around each closed geodesic. Second, determine the normal form from the spectrum of the laplacian. Third, determine the metric from the normal form. We will try to explain all three steps and to illustrate with simple examples such as surfaces...

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