General dispersion relation identities for electron states in crystals

Dispersion relation identities are sought, on the lines of those known and used in field theory and particle physics, which would govern the behaviour of electron states in a crystal, independent of the crystal's structure. It is shown that, in the case of a linear continuous periodic potential, a rigorous deduction is possible of relations exemplified by pi 2 N 2 (E)=E-w+2 Sigma n (n-1) integral (n)thgap (Imk (E')(E'-E) -1 )dE' where N(E) is the integrated state density and w the integrated well strength. For three dimensional crystals with no glide planes or screw axes, the problem is approached in the weak and tight binding approximations. It is shown that, within their intrinsic limits of accuracy, each approximation gives the 'subtracted' relation (with E 1 ,E 2 allowed) k 2 (E 1 )-k 2 (E 2 )-E 1 +E 2 = pi -1 integral - infinity infinity Imk 2 (E') ((E'-E 1 -1 -(E'-E 2 ) -1 )dE'. Here Imk 2 is to be calculated for a state attenuated in the direction of one of a specified set of reciprocal lattice vectors. These states are possible 'surface states', and it is noted that all similar relations will provide such a connection between surface states and body states.