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We prove dispersive and Strichartz estimates for Schr\"{o}dinger equations on normal real form symmetric spaces. These estimates apply to the well-posedness and scattering for the nonlinear Schr\"{o}dinger equations.

Analysis of PDEs · Mathematics 2019-10-17 Anestis Fotiadis , Effie Papageorgiou

We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties…

Analysis of PDEs · Mathematics 2023-02-14 Jean-Philippe Anker , Stefano Meda , Vittoria Pierfelice , Maria Vallarino , Hong-Wei Zhang

In this paper we focus on the validity of some fundamental estimates for time-degenerate Schr\"{o}dinger-type operators. On one hand we derive global homogeneous smoothing estimates for operators of any order by means of suitable comparison…

Analysis of PDEs · Mathematics 2024-02-19 Serena Federico , Michael Ruzhansky

In this work we prove a Strichartz estimate for the Schr\"odinger equation in the quasiperiodic setting. We also show a lower bound on the number of resonant frequency interactions in this situation.

Analysis of PDEs · Mathematics 2023-06-13 Friedrich Klaus

The primary objective in this paper is to give an answer to an open question posed by J. A. Barcel\'o, J. M. Bennett, A. Carbery, A. Ruiz and M. C. Vilela concerning the problem of determining the optimal range on $s\geq0$ and $p\geq1$ for…

Analysis of PDEs · Mathematics 2019-07-24 Youngwoo Koh , Ihyeok Seo

We prove global-in-time Strichartz estimates without loss of derivatives for the solution of the Schroedinger equation on a class of non-trapping asymptotically conic manifolds. We obtain estimates for the full set of admissible indices,…

Analysis of PDEs · Mathematics 2016-02-24 Andrew Hassell , Junyong Zhang

We obtain improved Strichartz estimates for solutions of the Schr\"odinger equation on compact manifolds with nonpositive sectional curvatures which are related to the classical universal results of Burq, G\'erard and Tzvetkov [11]. More…

Analysis of PDEs · Mathematics 2024-07-19 Xiaoqi Huang , Christopher D. Sogge

This paper proves Strichartz estimates for the Schrodinger Equation with a potential term and white noise dispersion in dimension $1$. We also explore dispersive estimates using previous results in the field.

Analysis of PDEs · Mathematics 2024-10-08 Abhinav Goel

We study the global-in-time Strichartz estimates for the Schr\"odinger equation on a class of scattering manifolds $X^{\circ}$. Let $\mathcal{L}_V=\Delta_g+V$ where $\Delta_g$ is the Beltrami-Laplace operator on the scattering manifold and…

Analysis of PDEs · Mathematics 2017-03-24 Junyong Zhang , Jiqiang Zheng

We give an optimal in mixed (anisotropic) Strichartz type Lebesgue space-time norm estimates for the solution of linear parabolic inhomogeneous initial problem, with are exact or exact up to multiplicative constant coefficient evaluation.

Analysis of PDEs · Mathematics 2014-01-07 E. Ostrovsky , L. Sirota

This note presents a new proof of the well-known Strichartz estimates for the Schr\"odinger equation in $2+1$ dimensions, building on ideas from our recent work \cite{MO}.

Classical Analysis and ODEs · Mathematics 2023-02-23 Camil Muscalu , Itamar Oliveira

We study the nonlinear Schr\"odinger equation associated with the sublaplacian L on the unit sphere $S^{2n+1}$ in $C^{n+1}$ equipped with its natural CR structure. We first prove Strichartz estimates with fractional loss of derivatives for…

Analysis of PDEs · Mathematics 2013-05-03 Valentina Casarino , Marco M. Peloso

We establish uniform Lipschitz estimates for second-order elliptic systems in divergence form with rapidly oscillating, almost-periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded…

Analysis of PDEs · Mathematics 2014-09-29 Scott N. Armstrong , Zhongwei Shen

We prove dispersive and Strichartz estimates for Schr\"o- dinger equations on a class of locally symmetric spaces {\Gamma}\X, where X = G/K is a symmetric space and {\Gamma} is a torsion free discrete sub- group of G. We deal with the cases…

Analysis of PDEs · Mathematics 2015-09-16 Anestis Fotiadis , Nikolaos Mandouvalos , Michel Marias

The purpose of this paper is to study the validity of global-in-time Strichartz estimates for the Schr\"odinger equation on $\mathbb{R}^n$, $n\ge3$, with the negative inverse-square potential $-\sigma|x|^{-2}$ in the critical case…

Analysis of PDEs · Mathematics 2017-05-09 Haruya Mizutani

In this note we obtain some Strichartz estimates for the Schr\"odinger equation associated to the twisted Laplacian on $\mathbb{C}^{n}\cong \mathbb{R}^{2n}$. The initial data will be considered in suitable Sobolev spaces associated to the…

Analysis of PDEs · Mathematics 2019-12-10 Duván Cardona

We prove Strichartz estimates for the Schroedinger operator $H = -\Delta + V(t,x)$ with time-periodic complex potentials $V$ belonging to the scaling-critical space $L^{n/2}_x L^\infty_t$ in dimensions $n \ge 3$. This is done directly from…

Analysis of PDEs · Mathematics 2007-11-03 Michael Goldberg

We prove sharp Strichartz estimates for the semi-classical Schrodinger equation on a compact manifold with smooth, strictly geodesically concave boundary. We deduce sharp (classical) Strichartz estimates for the Schrodinger equation outside…

Analysis of PDEs · Mathematics 2009-09-04 Oana Ivanovici

We prove inverse Strichartz theorems at $L^2$ regularity for a family of Schr\"{o}dinger evolutions in one space dimension. Prior results rely on spacetime Fourier analysis and are limited to the translation-invariant equation $i\partial_t…

Analysis of PDEs · Mathematics 2017-01-05 Casey Jao , Rowan Killip , Monica Visan

We consider the one-dimensional discrete Schr\"odinger operator $$ \bigl[H(x,\omega)\varphi\bigr](n)\equiv -\varphi(n-1)-\varphi(n+1) + V(x + n\omega)\varphi(n)\ , $$ $n \in \mathbb{Z}$, $x,\omega \in [0, 1]$ with real-analytic potential…

Spectral Theory · Mathematics 2018-09-26 Michael Goldstein , David Damanik , Wilhelm Schlag , Mircea Voda