Dirac-delta potentials on a lattice

Abstract

In this thesis, the Dirac delta potential on a one dimensional infinite lattice structure [2] and its band gaps [3] are reviewed. Then, Green’s function and regu larization & renormalization process for the bound states in two dimensions are men tioned [6–8]. The relation between the Green’s function and the renormalization of the interaction cofactor are shown. After that, the work of Albeverio on two dimensional lattice which shows that there is a solution of the Hamiltonian is reviewed [2]. In this model, the positions of Dirac delta potentials are in all lattice points in the one-to-one manner [10,11].Renormalization process is needed. Semi-relativisitic approach is used to look Dirac-delta potentials on infinite two dimensional lattice. Green’s function is found for the classic and semi-relativistic cases. Lastly, a new formula of interaction cofactor is found for the arbitrary shape of Dirac-delta potential in two dimensional lattice in the aspects of Quantum Mechanics. And, the Green’s function is written for this model. The formula shows us the existence of a solution of the Hamiltonian. Also, the three dimensional version of the formula is mentioned briefly. This formula is applied on three different patterns being two of them are line and the other pattern is circular. None of these examples do not need renormalization.