论文标题
连续模型的一般边界条件的形式主义
Formalism of general boundary conditions for continuum models
论文作者
论文摘要
由于其批量的汉密尔顿人的简单性，连续模型对约束国家的理论研究特别有吸引力。这条路径上的主要挑战是对边界的系统描述，该描述归结为确定适当的边界条件（BCS）。 BC是量子力学基本原理的结果：波函数的规范保护，这导致边界处的概率电流保存。 {\ em General Bcs}的概念是作为满足当前保存原则的所有可能BC的家庭。 Ahari，Ortiz和Seradjeh从最通用形式的一维哈密顿的当前保存原理中制定了总BC的系统推导程序。该过程基于电流的对角线化，并导致通用的``标准化''形式的通用BCS，通过单位矩阵以非冗余的一对一方式进行参数化。在这项工作中，我们对连续模型的一般边界条件的{\ em形式主义}进行了证实，阐明和扩展，详细介绍了许多重要的物理和数学点。我们从当前的保存原理中提供了一般BC的详细推导，并确定了它们在描述一个定义明确的边界的意义上可以接受的条件，该边界与自我伴侣（Hermitian）和仅对称操作员之间的微妙但至关重要的区别直接相关。我们提供了对普通BC的结构的自然物理解释，作为一个散射过程和基本的数学理由，即形式主义对于高于线性的动量秩序的哈密顿量明确定义。我们讨论了BCS一般的物理含义，并概述了形式主义的应用方案，特别是在拓扑系统中的结合状态研究。
Continuum models are particularly appealing for theoretical studies of bound states, due to simplicity of their bulk Hamiltonians. The main challenge on this path is a systematic description of the boundary, which comes down to determining proper boundary conditions (BCs). BCs are a consequence of the fundamental principle of quantum mechanics: norm conservation of the wave function, which leads to the conservation of the probability current at the boundary. The notion of {\em general BCs} arises, as a family of all possible BCs that satisfy the current-conservation principle. Ahari, Ortiz, and Seradjeh formulated a systematic derivation procedure of the general BCs from the current-conservation principle for the 1D Hamiltonian of the most general form. The procedure is based on the diagonalization of the current and leads to the universal ``standardized'' form of the general BCs, parameterized in a nonredundant one-to-one way by unitary matrices. In this work, we substantiate, elucidate, and expand this {\em formalism of general boundary conditions for continuum models}, addressing in detail a number of important physical and mathematical points. We provide a detailed derivation of the general BCs from the current-conservation principle and establish the conditions for when they are admissible in the sense that they describe a well-defined boundary, which is directly related to a subtle but crucial distinction between self-adjoint (hermitian) and only symmetric operators. We provide a natural physical interpretation of the structure of the general BCs as a scattering process and an essential mathematical justification that the formalism is well-defined for Hamiltonians of momentum order higher than linear. We discuss the physical meaning of the general BCs and outline the application schemes of the formalism, in particular, for the study of bound states in topological systems.
