$\mathrm{L}^{2}_{\alpha} $-Solutions for Nonlinear Schr\"odinger Equations Associated With The Weinstein Operator

TL;DR

This paper establishes the existence and uniqueness of global solutions for nonlinear Schr"odinger equations associated with Weinstein operators, expanding the understanding of such equations in weighted function spaces.

Contribution

It introduces the concept of $ ext{L}^2_eta$-solutions for nonlinear Schr"odinger equations linked to Weinstein operators, providing new results on solution existence and uniqueness.

Findings

Proved global existence of solutions in specified function spaces.

Established uniqueness of solutions under given initial conditions.

Extended analysis to Schr"odinger equations with Weinstein operators.

Abstract

In this paper, we study the Schr\"odinger equation associated with the Weinstein operators and we prove the existence and uniqueness of global solutions to Schr\"odinger-Weinstein equations in\\ $C\left(\left(-T_{\min }, T_{\max }\right) ; L_{\alpha}^{2}\left(\mathbb{R}^{d+1}_+\right)\right) \bigcap L_{l o c}^{q_{1}}\left(\left(-T_{\min }, T_{\max }\right) ; L_{\alpha}^{r_{1}}\left(\mathbb{R}^{d+1}_+\right)\right)$ \\ on initial condition in $L_{\alpha}^{2}\left(\mathbb{R}^{d+1}_+\right)$