Abstract: This paper introduces regularization methods and the concept of renormalization by analyzing the bound and scattering states of the two-dimensional Dirac delta function potential. First, it discusses the difficulties in solving the bound states of the Dirac delta function potential. Then, it establishes the relationship between the bound state energy values and theoretical coupling parameters through regularization. Next, based on this relationship, it analyzes the scattering cross-section of the potential and demonstrates its independence from the regularization process. Finally, a brief conclusion is provided.

Key words: Dirac function potential, quantum field theory, regularization, renormalization, functional analysis