Self-Adjoint extensions of the one-dimensional Schr\"odinger operator with symmetric potential

TL;DR

This paper explicitly characterizes the self-adjoint extensions of the one-dimensional Schrödinger operator with symmetric potential, linking boundary conditions to unitary matrices, and extends the analysis to non-symmetric potentials.

Contribution

It provides an explicit correspondence between self-adjoint domain extensions and boundary conditions, including non-symmetric potentials, clarifying the parametrization by unitary matrices.

Findings

Explicit boundary condition characterization for symmetric potentials

Connection between unitary matrices and boundary conditions

Extension of results to non-symmetric potentials

Abstract

We give an explicit correspondence between the domains of the self-adjoint extensions of a one-dimensional Schr\"odinger differential operator with symmetric real-valued potential and the boundary conditions the functions in the resulting domains must satisfy. As is well known, each self-adjoint extension is parametrized by a unitary matrix. We make the correspondence of this unitary matrix with the boundary conditions explicit, recovering the most familiar types of boundary conditions as special cases. We also demonstrate this correspondence implicitly for non-symmetric real-valued potential.