  Topological quantum phases of matter are characterized by an intimate
relationship between the Hamiltonian dynamics away from the edges and the
appearance of bound states localized at the edges of the system. Elucidating
this correspondence in the continuum formulation of topological phases, even in
the simplest case of a one-dimensional system, touches upon fundamental
concepts and methods in quantum mechanics that are not commonly discussed in
textbooks, in particular the self-adjoint extensions of a Hermitian operator.
We show how such topological bound states can be derived in a prototypical
one-dimensional system. Along the way, we provide a pedagogical exposition of
the self-adjoint extension method as well as the role of symmetries in
correctly formulating the continuum, field-theory description of topological
matter with boundaries. Moreover, we show that self-adjoint extensions can be
characterized generally in terms of a conserved local current associated with
the self-adjoint operator.
