Universal behavior of quantum Green's functions

Abstract

We consider a general one-particle Hamiltonian H = - Δ_r + u(r) defined in a d-dimensional domain. The object of interest is the time-independent Green function G_z(r,r&#39;) = < r | (z-H)^{-1} | r&#39; >. Recently, in one dimension (1D), the Green&#39;s function problem was solved explicitly in inverse form, with diagonal elements of Green&#39;s function as prescribed variables. The first aim of this paper is to extract from the 1D inverse solution such information about Green&#39;s function which cannot be deduced directly from its definition. Among others, this information involves universal, i.e. u(r)-independent, behavior of Green&#39;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&#39;s function for various shapes of the confining domain boundary.