Calculate the eigenenergy splitting present at the

Question

Exercise -

The eigenstates of electrons in a periodic potential are described by Bloch waves. Each Bloch factor has to obey the respective differential equation

Please prove -

(a) The Kramer's degeneracy (valid for same band index σ) ε_σ(k^→) = εσ(-k ^→), and

(b) The relation ε_σ'(k→ + K^→_h→) = ε_σ(k^→) and the property for the set of Bloch waves

(c) Calculate the eigenenergy splitting present at the Brillouin zone boundaries via the Bloch wave solutions valid in a case of a weak potential modulation felt by the electrons.