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Nikolaos Pattakos

Publications and source records attributed to Nikolaos Pattakos.

17 recordsLinked to original sources
arXiv2025

On the Excess Charge Problem of Atoms

This paper establishes new bounds on the maximum number of electrons $ N_c(Z) $ that an atom with nuclear charge $Z$ can bind. Specifically, we show that \begin{equation*} N_c(Z) < 1.1185Z + O(Z^{1/3}) \end{equation*} with an explicit bound on the lower order term $O(Z^{1/3})$. This result improves long--standing bounds by Lieb and Nam obtained in 1984, respectively 2012. Our bounds show the fundamental difference between fermionic and bosonic atoms for finite $Z$ since for bosonic atoms it is known that $\lim N_c(Z)/Z = t_c \approx 1.21$ in the limit of large nuclear charges $Z$.

math-ph↗
arXiv2019

Unconditional uniqueness of higher order nonlinear Schrödinger equations

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation with initial data $u_{0}\in X$, where $X\in\{M_{2,q}^{s}(\mathbb R), H^σ(\mathbb T), H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T)\}$ and $q\in[1,2]$, $s\geq0$, or $σ\geq0$, or $s_{2}\geq s_{1}\geq0$. Moreover, if $M_{2,q}^{s}(\mathbb R)\hookrightarrow L^{3}(\mathbb R)$, or if $σ\geq\frac16$ or if $s_{1}\geq\frac16$ and $s_{2}>\frac12$ we show that the Cauchy problem is unconditionally wellposed in $X$. Similar results hold true for all higher order nonlinear Schrödinger equations and mixed order NLS due to a factorization property of the corresponding phase factors. For the proof we employ the normal form reduction via the differentiation by parts technique and build upon our previous work

math.AP↗
arXiv2019

On the global wellposedness of the Klein-Gordon equation for initial data in modulation spaces

We prove global wellposedness of the Klein-Gordon equation with power nonlinearity $|u|^{α-1}u$, where $α\in\left[1,\frac{d}{d-2}\right]$, in dimension $d\geq3$ with initial data in $M_{p, p'}^{1}(\mathbb{R}^d)\times M_{p,p'}(\mathbb{R}^d)$ for $p$ sufficiently close to $2$. The proof is an application of the high-low method described by Bourgain [1] where the Klein-Gordon equation is studied in one dimension with cubic nonlinearity for initial data in Sobolev spaces.

math.AP↗
arXiv2019

On the global well-posedness of the quadratic NLS on $L^2(\mathbb{R}) + H^1(\mathbb{T})$

We study the one dimensional nonlinear Schrödinger equation with power nonlinearity $|u|^{α- 1} u$ for $α\in [1,5]$ and initial data $u_0 \in L^2(\mathbb{R}) + H^1(\mathbb{T})$. We show via Strichartz estimates that the Cauchy problem is locally well-posed. In the case of the quadratic nonlinearity ($α= 2$) we obtain global well-posedness in the space $C(\mathbb{R}, L^2(\mathbb R) + H^1(\mathbb T))$ via Gronwall's inequality.

math.AP↗
arXiv2019

Local well-posedness for the nonlinear Schrödinger equation in the intersection of modulation spaces $M_{p, q}^s(\mathbb{R}^d) \cap M_{\infty, 1}(\mathbb{R}^d)$

We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in Sugimoto (2011), of the intersection $M^s_{p,q}(\mathbb{R}^d) \cap M_{\infty, 1}(\mathbb{R}^d)$ for $d \in \mathbb{N}$, $p, q \in [1, \infty]$ and $s \geq 0$. We employ this algebra property to show the local well-posedness of the Cauchy problem for the cubic nonlinear Schrödinger equation in the above intersection. This improves Theorem 1.1 by Bényi and Okoudjou (2009), where only the case $q = 1$ is considered, and closes a gap in the literature. If $q > 1$ and $s > d \left(1 - \frac{1}{q}\right)$ or if $q = 1$ and $s \geq 0$ then $M^s_{p,q}(\mathbb{R}^d) \hookrightarrow M_{\infty, 1}(\mathbb{R}^d)$ and the above intersection is superfluous. For this case we also reobtain a Hölder-type inequality for modulation spaces.

math.AP↗
arXiv2019

The global Cauchy problem for the NLS with higher order anisotropic dispersion

We use a method developed by Strauss to obtain global wellposedness results in the mild sense for the small data Cauchy problem in modulation spaces $M_{p,q}^s(\mathbb{R}^d)$, where $q=1$ and $s\geq0$ or $q\in(1,\infty]$ and $s>\frac{d}{q'}$ for a nonlinear Schrödinger equation with higher order anisotropic dispersion and algebraic nonlinearities.

math.AP↗
arXiv2019

Knocking out teeth in one-dimensional periodic NLS

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in one dimension with initial data $u_{0}$ in $H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T), 0\leq s_{1}\leq s_{2}.$ In addition, we show that if $u_{0}\in H^{s}(\mathbb R)+H^{\frac12+ε}(\mathbb T)$ where $ε>0$ and $\frac16\leq s\leq\frac12$ the solution is unique in $H^{s}(\mathbb R)+H^{\frac12+ε}(\mathbb T).$ Our main tool is a normal form type reduction via the use of the differentiation by parts technique.

math.AP↗
arXiv2018

Nonlinear Schrödinger equation, differentiation by parts and modulation spaces

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space $M_{p,q}^{s}(\mathbb R)$ where $1\leq q\leq2$, $2\leq p<\frac{10q'}{q'+6}$ and $s\geq0$. Moreover, for either $1\leq q\leq\frac32, s\geq0$ and $2\leq p\leq 3$ or $\frac32 \frac23-\frac1{q}$ and $2\leq p\leq 3$ or $\frac{18}{11} \frac23-\frac1{q}$ and $2\leq p<\frac{10q'}{q'+6}$ we show that the Cauchy problem is unconditionally wellposed in $M_{p,q}^{s}(\mathbb R).$ This improves \cite{NP}, where the case $p=2$ was considered and the differentiation by parts technique was introduced to a problem with continuous Fourier variable. Here the same technique is used, but more delicate estimates are necessary for $p\neq2$.

math.AP↗
arXiv2018

NLS in the modulation space $M_{2,q}(\mathbb R)$

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in the modulation space $M_{2,q}^{s}(\mathbb R)$, $1\leq q\leq2$ and $s\geq0.$ In addition, for either $s\geq 0$ and $1\leq q\leq\frac32$ or $\frac32 \frac23-\frac1{q}$ we show that the Cauchy problem is unconditionally wellposed in $M_{2,q}^{s}(\mathbb R).$ It is done with the use of the differentiation by parts technique which had been previously used in the periodic setting.

math.AP↗
arXiv2017

On sharp bilinear Strichartz estimates of Ozawa-Tsutsumi type

We provide a comprehensive analysis of sharp bilinear estimates of Ozawa-Tsutsumi type for solutions u of the free Schrödinger equation, which give sharp control on $|u|^2$ in classical Sobolev spaces. In particular, we provide a generalization of their estimates in such a way that provides a unification with some sharp bilinear estimates proved by Carneiro and Planchon-Vega, via entirely different methods, by seeing them all as special cases of a one parameter family of sharp estimates. We show that the extremal functions are solutions of the Maxwell-Boltzmann functional equation and provide a new proof that this equation admits only Gaussian solutions. We also make a connection to certain sharp estimates on $u^2$ involving certain dispersive Sobolev norms.

math.AP↗
arXiv2016

On existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space $M_{p,q}(\mathbb R)$

We prove global existence for the one-dimensional cubic non-linear Schrödinger equation in modulation spaces $M_{p,p'}$ for $p$ sufficiently close to $2$. In contrast to known results, our result requires no smallness condition on initial data. The proof adapts a splitting method inspired by work of Vargas-Vega and Hyakuna-Tsutsumi to the modulation space setting and exploits polynomial growth of the free Schrödinger group on modulation spaces.

math.AP↗
arXiv2015

A dyadic analysis approach to the problem of continuity of weighted estimates with respect to the $A_p$ characteristic

This paper presents a new proof of the results regarding the continuity of weighted estimates with respect to the characteristic of the weight. Here we first prove the result in the dyadic case which is "easier" and then by the use of the Bellman function technique we pass to the continuous setting which is harder in general. To be more precise, as far as we know, this passage from the Martingale transform to the Hilbert transform described in this note is new.

math.CA↗
arXiv2013

Matrix weights, Littlewood Paley inequalities and the Riesz transforms

We discuss weighted estimates for the squares of the Riesz transforms, R^{2}, on L^{2}(W) where W is a matrix A2 weight. We prove that if W is close to the Identity matrix Id, then the operator norm of R^{2} is close to its unweighted norm on L^{2} which is one. This is done by the use of the Bellman function technique.

math.CA↗
arXiv2012

Continuity of weighted estimates for sublinear operators

In this note we prove that if a sublinear operator T satisfies a certain weighted estimate in the $L^{p}(w)$ space for all $w\in A_{p}$, $1<p<+\infty$, then the operator norm of T on $L^{p}(w)$ is a continuous function of the weight $w$, with respect to a certain metric $d_{*}$ on $A_{p}$. This, generalizes a previous result on the same subject for linear operators.

math.CA↗
arXiv2012

Some remarks on extrapolation with "flat" weights

We prove an extrapolation result for general operators under some weak assumptions on the boundedness of the operator. In particular, we show that if the operator is weakly bounded on some L^{p_{0}}(w), for all "flat" weights, w in A_{p_{0}}, 1 < p_{0} <\infty, then for p in some small neighborhood around p_{0}, and all "flat" A_{p} weights, w, the operator is weakly bounded on L^{p}(w), and as a result we get strong type estimates for the operator.

math.CA↗
arXiv2011

The Muckenhoupt $A_\infty$ class as a metric space and continuity of weighted estimates

We show how the $A_\infty$ class of weights can be considered as a metric space. As far as we know this is the first time that a metric d is considered on this set. We use this metric to generalize the results obtained in [9]. Namely, we show that for any Calderon- Zygmund operator T and an $A_p$, 1 < p < 1, weight $w_0$, the operator norm of T in $L^{p}(w)$ converge to the operator norm of T in L^{p}(w_{0})$ as d(w;w_0) goes to 0. We also find the rate of this convergence and prove that is sharp.

math.CA↗