Time-independent scalar potentials

Consider a particle in a time-independent scalar potential.
The Schroedinger equation for such a particle is iħ∂Ψ(r,t)/∂t = -(ħ2/(2m))∇2Ψ(r,t) + U(r)Ψ(r,t).
The wave function is a function of position r and time t.

Is separation of variables possible?
Assume solutions of the form Ψ(r,t) = Φ(r)ζ(t) exist.   
Inserting Ψ(r,t) = Φ(r)ζ(t) into the Schroedinger equation we obtain
(iħ/ζ(t)) ∂ζ(t)/∂t  = (1/Φ(r))[-(ħ2/(2m))∇2Φ(r) ] + U(r)Φ(r).
The left hand side of the equation is a function of t, independent of r, and the right hand side is a function of r, independent of t.  Both sides must therefore be equal to a constant, which we call E.  The left hand side then yields

iħ∂ζ(t))/∂t = Eζ(t).   ζ(t) = A exp(-iEt/ħ) = A exp(-iωt).

Here ω is the frequency of the wave function solution, and the de Broglie relation identifies E as the energy of the particle. 
The right hand side yields H Φ(r) = E Φ(r), with  H = -(ħ2/(2m))∇2 + U(r) being a differential operator, or
2Φ(r) + (2m/ħ2)(E - U(r))Φ(r) = 0.

If U = U(r), then solutions of the type Φ(r)ζ(t) exist.  They are called stationary solutions or stationary states.  They correspond to states in which the particle has a well defined energy E = ħω. 
Φ(r) is a solution of HΦ(r) = EΦ(r), the time-independent Schroedinger equation.  It is an energy eigenfunction, since the time-independent Schroedinger equation is an eigenvalue equation of the linear operator H.  The allowed energies are called the eigenvalues of H.

If solutions Φn(r) can be found, for different values of En, then the linear combination 
Ψ(r,t) = ∑n cnΦn(r)exp(-iEnt/ħ)
is also a solution of the Schroedinger equation, because the Schroedinger equation is a linear equation.
However, such a linear combination does not represent a particle with a well defined energy.