The discussion centers on the representation of periodic potentials in the form v(r)=Ʃf(r-R), where the sum is over lattice vectors R. Participants confirm that this formulation is valid under the condition that the potential v is periodic, satisfying v(r+R) = v(r) for all lattice vectors R. The conversation also touches on the implications of periodic boundary conditions and the relationship between the potential and the lattice structure. A specific method to derive this representation involves setting f(r) = v(r)/N_R, where N_R is the total number of lattice points.
PREREQUISITESPhysicists, materials scientists, and computational researchers interested in solid-state physics and the mathematical modeling of periodic systems.
Could you please prove it or give me a reference which has proved it?cgk said:yes, if the potential v is periodic, then it can be written in this form. This is the most general form of a function v(r) which fulfills v(r+R) = v(r) for all lattice vectors R (i.e., which is periodic).