| 12/11/20 | Seminario | 14:00 | 15:00 | | Claudianor Alves | Università Federale di Campina Grande (Brasile) | Super-critical Neumann problems on unbounded domains
In this paper, by making use of a new variational principle, we prove existence of nontrivial solutions for two different types of semilinear problems with Neumann boundary conditions in unbounded domains. Namely, we study elliptic equations and Hamiltonian systems on the unbounded domain $Omega=R^{m} imes B_r$ where $B_r$ is a ball centered at the origin with radius $r$ in $mathbb{R}^{n}$. Our proofs consist of several new and novel ideas that can be used in broader contexts. This is a joint work with Abbas Moameni that was accepted for publication in Nonlinearity.
N.B.: this talk is part of the activity of the MIUR Excellence Department Project CUP E83C18000100006.
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| 05/11/20 | Seminario | 16:30 | 17:30 | | Adrian Perez Bustamante | Georgia Institute of Technology | Gevrey estimates and domains of analyticity for asymptotic expansions of tori in weakly dissipative systems
We consider the problem of following quasi-periodic tori in perturbations of some Hamiltonian systems which involve friction and external forcing. In a first goal, we use different numerical methods (Pade approximants, Newton continuation till boundary) to obtain numerically the domain of convergence. We also study the properties of the asymptotic series of the solution. In a second goal, we study rigorously the (divergent) series of formal expansions of the torus obtained using Lindstedt method. We show that, for some systems in the literature, the series is Gevrey. We hope that the method can be of independent interest: we develop KAM estimates for the divergent series. In contrast with the regular KAM method, we lose control of all the domains, so that there is no convergence, but we can generate enough control to show that the series is Gevrey.
This is joint work with R. Calleja and R. de la Llave.
This activity is made in collaboration with the Departments of Mathematics of the Universities of Milano, Padova, Pisa and Roma Tor Vergata (Excellence Department project MATH@TOV). |
| 27/10/20 | Seminario | 14:00 | 15:00 | 1201 Dal Passo | Erik Tonni | SISSA |
Modular Hamiltonians for the massless Dirac field in the presence of a boundary or of a defect
The reduced density matrix of a spatial subsystem can be written as the exponential of the modular Hamiltonian, hence this operator contains a lot of information about the entanglement of the corresponding spatial bipartition. First we consider the massless Dirac field on the half-line, imposing the most general boundary conditions that ensure the global energy conservation. This leads to two inequivalent phases where either the vector or the axial symmetry is preserved. In these two phases, we discuss the analytic expressions for the modular Hamiltonians of an interval on the half-line when the system is in its ground state, for the corresponding modular flows of the Dirac field and for the corresponding modular correlators. The method allows to obtain analytic expressions also for the modular Hamiltonians, the modular flows and the modular correlators for two disjoint equal intervals at the same distance from a point-like defect characterised by a unitary scattering matrix, that allows both reflection and transmission.
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