  Transposed Poisson structures on the Schr\"{o}dinger algebra in
$(n+1)$-dimensional space-time of Schr\"{o}dinger Lie groups are described. It
was proven that the Schr\"{o}dinger algebra $\mathcal{S}_{n}$ in case of $n\neq
2$ does not have non-trivial $\frac{1}{2}$-derivations and as it follows it
does not admit non-trivial transposed Poisson structures. All
$\frac{1}{2}$-derivations and transposed Poisson structures for the algebra
$\mathcal{S}_{2}$ are obtained. Also, we proved that the Schr\"{o}dinger
algebra $\mathcal{S}_{2}$ admits a non-trivial ${\rm Hom}$-Lie structure.
