On Green's functions for Hamiltonians with potentials possessing singularity at the origin: application to the zero-range potential formalism

TL;DR

This paper analyzes the short-range behavior of Green's functions for Hamiltonians with inverse power singular potentials, extending the zero-range potential formalism to include such singularities.

Contribution

It provides a detailed asymptotic analysis of Green's functions with inverse power singularities and extends the zero-range potential formalism to these cases.

Findings

For weaker than Coulomb singularities, Green's function has standard singularity.

For $r^{- ho}$ potentials with $1 extless ho extless2$, an additional singularity appears.

Logarithmic singularity occurs for Coulomb ($ ho=1$) potential.

Abstract

We evaluate the short-range asymptotic behavior of Green's function for a Hamiltonian when its potential energy part has an inverse power singularity at the origin. The analytically solvable case of sharply screened Coulomb potential is considered firstly. For this potential the additional logarithmic singular term has been found in the short-range asymptote of the Green function as in the case of the pure Coulomb potential. The case of a short-range potential of an arbitrary form with inverse power singularity is treated on the basis of the integral Lippmann-Schwinger equation. It is shown that, if the singularity is weaker than the Coulomb one, the Green function has only standard singularity. For the case of $r^{-\rho}$ singularity of the potential with $1\le\rho<2$ the additional singularity in the asymptotic behavior of the Green function appears. In the case of $\rho=1$ the additional logarithmic singularity has the same form as in the case of the pure Coulomb potential. In the case of $1<\rho<2$ the additional singularity of the Green function has the form of the polar singularity $r^{-\rho+1}$. These results are applied for extending the zero-range potential formalism on Hamiltonians with singular potentials.