for the solution to the time-independent Schrödinger's equation with a period potential, [tex]V(x)=V(x+a)[/tex] one has: [tex]\psi(x)=Af(x)+Bg(x)[/tex] and [tex]\psi(x+a)=A'f(x)+B'g(x)[/tex] the coefficients are related by a matrix equation, let [tex]v=[A, B]^T[/tex] [tex]v'=[A', B']^T[/tex] then [tex]v=Kv'[/tex] where K is some non-singular matrix. hence we can choose a solution where v is the eigenvector to the matrix K, more specifically, [tex]v'=Kv=\lambda v[/tex] and thus one can find solutions to the Schrödinger's equation that satisfy: [tex]f(x+a)=\lambda f(x)[/tex] (1) from there, one can show that there are band structures for the allowed energies. However, how does one prove that these are the ONLY physical solution to the Schroedinger's equation? I mean without the constrain of (1), we can possibly have other wave functions with different energies (in the forbidden zone)? am I missing something?