量子格林函数的普适行为
Universal behavior of quantum Green's functions
摘要：
本文考虑一般单粒子哈密顿量H = - Δ r + u（r）定义在d维区域上。感兴趣的对象是与时间无关的格林函数G_z（r，r '）=|（z-H）^{-1}| r' >。最近，在一维（1D）中，格林函数问题以逆形式显式求解，其中格林函数的对角元素作为规定变量。本文的第一个目的是从一维逆解中提取格林函数的信息，这些信息不能直接从格林函数的定义中推导出来。其中，该信息涉及通用的，即u（r）独立的，靠近域边界的格林函数的行为。第二个目的是将逆形式主义扩展到更高的维度，特别是3D，并推导出各种形状的限制域边界的格林函数的通用形式。
英文摘要：
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') = < 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.
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