Noticias en español
Virtual levels and virtual states of operators in Banach spaces.
Abstract: Virtual levels admit several equivalent characterizations: (1) there are corresponding eigenstates from L^2 or a space “slightly weaker” than L^2; (2) there is no limiting absorption principle in the vicinity of a virtual level (e.g. no weights such that the “sandwiched” resolvent remains uniformly bounded); (3) an arbitrarily small perturbation can produce an eigenvalue. We develop a general approach to virtual levels in Banach spaces and provide applications to Schroedinger operators with non selfadjoint potentials and in any...
Read MoreAnisotropic harmonic maps and Ginzburg-Landau type relaxation.
Abstract: Consider maps $u:R^n\to R^k$ with values constrained in a fixed submanifold, and minimizing (locally) the energy $E(u)=\int W(\nablau)$. Here $W$ is a positive definite quadratic form on matrices. Compared to the isotropic case $W(\nabla u)=|\nabla u|^2$ (harmonic maps) this may look like a harmless generalization, but the regularity theory for general $W$’s is widely open. I will explain why, and describe results with Andres Contreras on a relaxed problem, where the manifold-valued constraint is replaced by an integral penalization.
Read MoreEigenvalue splitting of polynomial order for a system of Schrödinger operators with energy-level crossing.
In this talk I will present recent results on the spectral analysis of a 1D semiclassical system of coupled Schrödinger operators each of which has a simple well potential. Such systems arise as important models in the Born-Oppenheimer approximation of molecular dynamics. Focusing on the cases where the two underlying classical periodic trajectories cross to each other which imply an energy-level crossing, we give Bohr-Sommerfeld type quantization rules for the eigenvalues of the system in both tangential and transversal crossing cases. Our main results consists in the eigenvalue splitting...
Read MoreSolitons and multilinear Harmonic Analysis with potentials.
Abstract: We study some classes of nonlinear dispersive PDEs with potentials in both 1 and 3 dimensions, motivated by questions on the stability of solitons and topological solitons. Our approach is based on the so-called “distorted Fourier transform” adapted to Schrodinger operators, and the development of multilinear harmonic analysis in this setting. This approach allows us to treat low power nonlinearities and describe the global space-time behavior of solutions, also by capturing singularities in frequency space which may arise due to various coherent phenomena. Several...
Read MoreElapsed time neural assemblies and learning processes for weak interconnections.
Abstract: Modeling neural networks is an interesting problem from both mathematical and neuroscience point of view. In particular, evolution equations describing neural assemblies derived from stochastic processes and microscopic models have become a very active area in recent years. In this context we study a neural network via time-elapsed dynamics with learning, where neurons are described by their position in brain cortex and the elapsed time since last discharge. The learning process is modeled via the change in the interconnections among neurons which defines a learning rule. A typical...
Read MoreStrongly interacting multi-solitons and asymptotically static nonlinear waves.
Abstract: A generic global solution to a nonlinear wave equation exhibits oscillatory behavior, which is reflected in the fact that its kinetic energy does not tend to zero in infinite time. However, there are special solutions whose kinetic energy converges to zero and we call such solutions asymptotically stationary. These play an important role in the description of the phase portrait, whether they are stable or not. Familiar examples include stationary solutions (corresponding to critical points of the potential energy) or their stable manifolds. In this talk, we discuss asymptotically...
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