Ok I think I am able to calculate the eigenvalues of H with the recursion method 
 for the special case k=0 and psinc(1) (diagonal element, hermitian Hamiltonian).
 The eigenvalues do coincide with those of the CG method to within 1 meV after
 about 150 iterations.

 I obtained the eigenvalues by diagonalizing the tridiagonal matrix obtained from
 the recursion coefficients. Some non0degenerate eigenvalues appear multiple times,
 I don't know at this stage whether this is a problem for the computation of the
 Green's function or not. i would say no, since in the recursion methods there is
 no assumption about the multiplicity of the calculated spectrum.

 The Green's function has poles at SOME OF the eigenvalues, the strenght is nonzero
 when the initial state has components on that eigenstate.

 Now I need to try the nondiagonal elements, and the nonzero k vectors.

 *** TO CHECK THE GREEN'S FUNCTION: Test directly vs sum over empty states ***