  In this paper, we study the extremal problem for the Strichartz inequality
for the Schr\"{o}dinger equation on $\mathbb{R}^2$. We show that the solutions
to the associated Euler-Lagrange equation are exponentially decaying in the
Fourier space and thus can be extended to be complex analytic. Consequently we
provide a new proof to the characterization of the extremal functions: the only
extremals are Gaussian functions, which was investigated previously by Foschi
and Hundertmark-Zharnitsky.
