Infinitesmal deformations and central extensions of the $n$-th Schrödinger algebra

In this paper we study the second cohomology space for the $n$-th Schrödinger algebra ${\mathfrak{s}\mathfrak{c}\mathfrak{h}}_n$. We prove that the second cohomology space is vanishing for $n\ge 3$ and show that $\dim (H^2({\mathfrak{s}\mathfrak{c}\mathfrak{h}}_2,{\mathfrak{s}\mathfrak{c}\mathfrak{h}}_2))=2.$ Moreover, we investigate the factor algebra of the Schrödinger algebra by its center and determine the second cohomology group of this factor algebra.