A purely parabolic band corresponds to an independent particle in a constant external potential (technically, in a large box and subject to the periodic boundary condition), since in such potential the dispersion of the energy eigenvalue corresponding to a given wave vector k is determined solely by the kinetic-energy operator. The non-parabolicity of the energy dispersion is therefore a consequence of the external potential not being a constant. In such case, a plane wave is no longer an eigenfunction of the Hamiltonian so that the underlying time-independent Schrödinger equation will have to be solved, using a variety of techniques. This is done for instance by choosing an appropriate set of one-particle basis functions and calculating the matrix representation of the Hamiltonian operator with respect to this basis, and diagonalising the latter matrix. For details, consult any good book on quantum mechanics or solid state physics, such as Solid State Physics by Ashcroft and Mermin. For orientation, consider the one-dimensional Krönig-Penney model.

https://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice

A purely parabolic band corresponds to an independent particle in a constant external potential (technically, in a large box and subject to the periodic boundary condition), since in such potential the dispersion of the energy eigenvalue corresponding to a given wave vector k is determined solely by the kinetic-energy operator. The non-parabolicity of the energy dispersion is therefore a consequence of the external potential not being a constant. In such case, a plane wave is no longer an eigenfunction of the Hamiltonian so that the underlying time-independent Schrödinger equation will have to be solved, using a variety of techniques. This is done for instance by choosing an appropriate set of one-particle basis functions and calculating the matrix representation of the Hamiltonian operator with respect to this basis, and diagonalising the latter matrix. For details, consult any good book on quantum mechanics or solid state physics, such as Solid State Physics by Ashcroft and Mermin. For orientation, consider the one-dimensional Krönig-Penney model.

https://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice