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L. Vega

Publications and source records attributed to L. Vega.

17 recordsLinked to original sources
arXiv2024

On the fractional Sch\"odinger equation with variable coefficients

We study the initial value problem (IVP) associated to the semi-linear fractional Sch\"odinger equation with variable coefficients. We deduce several properties of the anisotropic fractional elliptic operator modelling the dispersion relation and use them to establish the local well-posedness for the corresponding IVP. Also, we obtain unique continuation results concerning the solutions of this problem. These are consequences of uniqueness properties that we prove for the fractional elliptic operator with variable coefficients

math.AP↗
arXiv2022

The Frisch--Parisi formalism for fluctuations of the Schrödinger equation

We consider the solution of the Schrödinger equation $u$ in $\mathbb{R}$ when the initial datum tends to the Dirac comb. Let $h_{\text{p}, δ}(t)$ be the fluctuations in time of $\int\lvert x\rvert^{2δ}\lvert u(x,t)\rvert^2\,dx$, for $0 < δ< 1$, after removing a smooth background. We prove that the Frisch--Parisi formalism holds for $H_δ(t) = \int_{[0,t]}h_{\text{p}, δ}(2s)\,ds$, which is morally a simplification of the Riemann's non-differentiable curve $R$. Our motivation is to understand the evolution of the vortex filament equation of polygonal filaments, which are related to $R$.

math.AP↗
arXiv2015

Mapping optically variable quasars towards the galactic plane

We present preliminary results of the CIDA Equatorial Variability Survey (CEVS), looking for quasar (hereafter QSO) candidates near the Galactic plane. The CEVS contains photometric data from extended and adjacent regions of the Milky Way disk ($\sim$ 500 sq. deg.). In this work 2.5 square degrees with moderately high temporal sampling in the CEVS were analyzed. The selection of QSO candidates was based on the study of intrinsic optical photometric variability of 14,719 light curves. We studied samples defined by cuts in the variability index (Vindex $>$ 66.5), periodicity index (Q $>$ 2), and the distribution of these sources in the plane (AT , $γ$), using a slight modification of the first-order of the structure function for the temporal sampling of the survey. Finally, 288 sources were selected as QSO candidates. The results shown in this work are a first attempt to develop a robust method to detect QSO towards the Galactic plane in the era of massive surveys such as VISTA and Gaia.

astro-ph.GA↗
arXiv2015

Hardy Uncertainty Principle, Convexity and Parabolic Evolutions

We give a new proof of the $L^2$ version of Hardy's uncertainty principle based on calculus and on its dynamical version for the heat equation. The reasonings rely on new log-convexity properties and the derivation of optimal Gaussian decay bounds for solutions to the heat equation with Gaussian decay at a future time. We extend the result to heat equations with lower order variable coefficient.

math.AP↗
arXiv2010

The Hardy Uncertainty Principle Revisited

We give a real-variable proof of the Hardy uncertainty principle. The method is based on energy estimates for evolutions with positive viscosity, convexity properties of free waves with Gaussian decay at two different times, elliptic $L^2$-estimates and the invertibility of the Fourier transform on $L^2(\Rn)$ and $\mathcal S'(\Rn)$.

math.AP↗
arXiv2009

Uncertainty Principle of Morgan type and Schrödinger Evolutions

We prove unique continuation properties for solutions of evolution Schrödinger equation with time dependent potentials. In the case of the free solution these correspond to uncertainly principles referred to as being of Morgan type. As an application of our method we also obtain results concerning the possible concentration profiles of solutions of semi-linear Schrödinger equations.

math.AP↗
arXiv2009

The sharp Hardy Uncertainty Principle for Schödinger evolutions

We give a new proof of Hardy's uncertainty principle, up to the end-point case, which is only based on calculus. The method allows us to extend Hardy's uncertainty principle to Schrödinger equations with non-constant coefficients. We also deduce optimal Gaussian decay bounds for solutions to these Schrödinger equations.

math.AP↗
arXiv2008

Weak Dispersive estimates for Schrödinger equations with long range potentials

We prove some local smoothing estimates for the Schrödinger initial value problem with data in $L^2(\mathbb{R}^d)$, $d \geq 2$ and a general class of potentials. In the repulsive setting we have to assume just a power like decay $(1+|x|)^{-γ}$ for some $γ>0$. Also attractive perturbations are considered. The estimates hold for all time and as a consequence a weak dispersion of the solution is obtained. The proofs are based on similar estimates for the corresponding stationary Helmholtz equation and Kato H-smooth theory.

math.AP↗
arXiv2008

Hardy's Uncertainty Principle, Convexity and Schrödinger Evolutions

We prove the logarithmic convexity of certain quantities, which measure the quadratic exponential decay at infinity and within two characteristic hyperplanes of solutions of Schrödinger evolutions. As a consequence we obtain some uniqueness results that generalize (a weak form of) Hardy's version of the uncertainty principle. We also obtain corresponding results for heat evolutions.

math.AP↗
arXiv2005

On unique continuation of solutions of Schrödinger equations

We study uniqueness properties of solutions of Schrödinger equations. The aim is to obtain sufficient conditions on the decay behavior of the difference of two solution $u_1-u_2$ of the equation at two different times $t_0=0$ and $t_1=1$ which guarantee the uniqueness of the solution, i.e. that $u_1\equiv u_2$.

math.AP↗
arXiv2005

Variable coefficient Schrödinger flows for ultrahyperbolic operators

In this paper we study the local solvability of nonlinear Schrödinger equations of the form $$\p_t u = i {\cal L}(x) u + \vec b_1(x)\cdot \nabla_x u + \vec b_2(x)\cdot \nabla_x \bar u + c_1(x)u+c_2(x)\bar u +P(u,\bar u,\nabla_x u, \nabla_x\bar u), where $x\in\mathbb R^n$, $t>0$, $\displaystyle{\cal L}(x) = -\sum_{j,k=1}^n\p_{x_j}(a_{jk}(x)\p_{x_k})$, $A(x)=(a_{jk}(x))_{j,k=1,..,n}$ is a real, symmetric and nondegenerate variable coefficient matrix, and $P$ is a polynomial with no linear or constant terms. Equations of the form described in with $A(x)$ merely invertible as opposed to positive definite arise in connection with water wave problems, and in higher dimensions as completely integrable models. Under appropriate assumptions on the coefficients we shall show that the associated initial value problem is local well posed.

math.AP↗
arXiv2003

An analytical proof of Hardy-like inequalities related to the Dirac operator

We prove some sharp Hardy type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac-Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity.

math-ph↗