Funções de Green da equação de Schrödinger no intervalo com condições de contorno auto-adjuntas arbitrárias

Abstract

This work aims to determine Green functions for the Schr¨odinger equation in the interval with arbitrary self-adjunct boundary conditions. First, von Neumann theory of self-adjoining extensions of symmetric operators was used to determine the existence of a quadri-parametric family of self-adjoining extensions of Schr¨odinger Hamiltonians. AIM formalism (Asorey, Ibort, Marmo) was used to determine the explicit shape of boundary conditions and the construction of the family of self-adjunct Hamiltonians. The scheme of proof of completeness for the sets of self-functions was presented, which is a crucial property for the construction of Green functions. Using the boundary conditions determined by the AIM formalism, Green functions were determined for the Schr¨odinger equation with arbitrary self-adjunct boundary conditions. As an application of these Green functions, the resolving function for this case was found and its veracity was verified by determining the Fourier series by the sets of auto functions of the Schr¨odinger equation for some frequently used cases.