Suppose $\phi(t)$ is a characteristic function of a random variable $X$. It satisfies $$\phi(t)=\phi^2\left(\frac{t}{\sqrt{2}}\right)$$
Further we know $\phi \in C^2(R)$.
Can we derive $\phi(t)$ is a characteristic function of a Gaussian random variable?

First since $\phi'(t) = \sqrt{2} \phi\left(\frac{t}{\sqrt{2}}\right) \phi'\left(\frac{t}{\sqrt{2}}\right)$, $\phi'(0)=0$, thus $E[X]=0$. For $X \sim N(0, \sigma^2)$, its characteristic function exactly satisfies that equation. But I'm stuck to prove the other side. Thanks for any help!