Universal behavior of quantum Green's functions

TL;DR

This paper investigates the universal, boundary-independent behavior of quantum Green's functions in various dimensions, extending inverse solution methods from 1D to 3D to understand their fundamental properties.

Contribution

It introduces a method to extract universal Green's function behavior near boundaries and extends inverse formalism to higher dimensions, including 3D.

Findings

Universal boundary behavior of Green's functions identified

Inverse formalism extended from 1D to 3D

Explicit forms derived for various domain shapes

Abstract

We consider a general one-particle Hamiltonian H = - \Delta_r + u(r) defined in a d-dimensional domain. The object of interest is the time-independent Green function G_z(r,r') = < r | (z-H)^{-1} | r' >. Recently, in one dimension (1D), the Green's function problem was solved explicitly in inverse form, with diagonal elements of Green's function as prescribed variables. The first aim of this paper is to extract from the 1D inverse solution such information about Green's function which cannot be deduced directly from its definition. Among others, this information involves universal, i.e. u(r)-independent, behavior of Green's function close to the domain boundary. The second aim is to extend the inverse formalism to higher dimensions, especially to 3D, and to derive the universal form of Green's function for various shapes of the confining domain boundary.