Are there closed solution for the wave function of schroedinger equation?
i mean solutions in the form $ \Psi (x,t)= f(x-t,y,z,t) $
that are not given by infinite series.
For example for the 1+1D wave function there is the closed d'Alambert solution
$$ f(x-vt)+ f(x+vt) $$ where $v$ is the speed of the propagation of the wave.
Any "nice" square-normalizable function $f(\textbf{x})$ can be a solution to the time-dependent Schrödinger equation at some point in time. To see this, consider the case of a time-independent Hamiltonian, and define the function
$$f(\textbf{x},t)\equiv e^{-iHt/\hbar}f(\textbf{x}).$$
It is clear from the functional form that $i\hbar\partial f(\textbf{x},t)/\partial t=Hf(\textbf{x},t)$, so that it is a solution to the Schrödinger Equation. If the Hamiltonian is time-dependent, then we can use Dyson's trick and define
$$f(\textbf{x},t)\equiv\text{Texp}\left[-\frac{i}{\hbar}\int_{0}^{t}H(t')\mathrm{d}t'\right]f(\textbf{x}),$$
where $\text{Texp}$ is the time-ordered exponential, the details of which I won't go into.
This is about as closed form as a solution to the Schrödinger equation gets, until you specify a potential.
