$\mathrm{L}^{2}_α $-Solutions for Nonlinear Schrödinger Equations Associated With The Weinstein Operator

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 $L_{\alpha}^{2}\left(\mathbb{R}^{d+1}_+\right)$