# Schrodinger Equation and wavefunctions

not all functions are wavefunctions. For functions to be wavefunctions they have to obey a series of "rules". Now, my question is:

there are many functions, which obey these rules which aren't eigenfunctions of the hamiltonian, thereby meaning that they don't obey the Schrodinger Equation. Can systems described by these kind of wavefunctions exist or is it, as I think it is, not physically possible??