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The Frisch–Parisi formalism for fluctuations of the Schrödinger equation
(2022)
We consider the solution of the Schrödinger equation $u$ in $\mathbb{R}$ when the initial datum tends to the Dirac comb. Let $h_{\text{p}, \delta}(t)$ be the fluctuations in time of $\int\lvert x \rvert^{2\delta}\lvert ...
Static and Dynamical, Fractional Uncertainty Principles
(2021-03)
We study the process of dispersion of low-regularity solutions to the Schrödinger equation using fractional weights (observables). We give another proof of the uncertainty principle for fractional weights and use it to get ...
Unbounded growth of the energy density associated to the Schrödinger map and the binormal flow
(2021)
We consider the binormal flow equation, which is a model for the dynamics
of vortex filaments in Euler equations. Geometrically it is a flow of curves in three dimensions, explicitly connected to the 1-D Schr¨odinger map ...
On the one dimensional cubic NLS in a critical space
(2021)
In this note we study the initial value problem in a critical space for the one dimensional Schr¨odinger equation with a
cubic non-linearity and under some smallness conditions. In particular the initial data is given by ...
Eigenvalue Curves for Generalized MIT Bag Models
(2021)
We study spectral properties of Dirac operators on bounded domains Ω ⊂ R
3 with
boundary conditions of electrostatic and Lorentz scalar type and which depend on a parameter
τ ∈ R; the case τ = 0 corresponds to the MIT ...
On the improvement of the Hardy inequality due to singular magnetic fields
(2020-09-01)
We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type ...
On the unique continuation of solutions to non-local non-linear dispersive equations
(2020-08-02)
We prove unique continuation properties of solutions to a large class of nonlinear, non-local dispersive equations. The goal is to show that if (Formula presented.) are two suitable solutions of the equation defined in ...
Riemann's non-differentiable function and the binormal curvature flow
(2020-07-14)
We make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object ...
Vortex Filament Equation for a regular polygon in the hyperbolic plane
(2020-07-09)
The aim of this article is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and ...
On the energy of critical solutions of the binormal flow
(2020-07-02)
The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic ...