Formalism of general boundary conditions for continuum models

TL;DR

This paper develops a comprehensive formalism for boundary conditions in continuum quantum models, ensuring norm conservation and applicability to complex Hamiltonians, with implications for topological bound states.

Contribution

It systematically derives and expands the general boundary condition formalism, clarifying physical and mathematical aspects, including higher-order Hamiltonians and boundary scattering interpretations.

Findings

Derived conditions for admissible boundary conditions

Provided a physical interpretation as scattering processes

Extended formalism to higher-order Hamiltonians

Abstract

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.