Problem: Density of States in Superconductors

         

In the BCS theory of superconductivity the excitation spectrum of the single-electron states is

where is the electron energy in the absence of superconductivity and is the Fermi energy. With a simplified interaction between the electrons, one obtains . Similar expression is obtained for other systems where interactions lead to nonmetallic ground state, like charge density waves, spin density waves, and so on. Calculate the DOS assuming that is not close to a van Hove singularity and .