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

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.