  This paper proposes a classification of elliptic (pseudo-)differential
Hamiltonians describing topological insulators and superconductors in Euclidean
space by means of domain walls. Augmenting a given Hamiltonian by one or
several domain walls results in confinement that naturally yields a Fredholm
operator, whose index is taken as the topological charge of the system. A
Fedosov-H\"ormander formula implementing in Euclidean spaces an Atiyah-Singer
index theorem allows for an explicit computation of the index in terms of the
symbol of the Fredholm operator. For Hamiltonians admitting an appropriate
decomposition in a Clifford algebra, the index is given by the easily
computable degree of a naturally associated map.
  A practically important property of topological insulators is the asymmetric
transport observed along one-dimensional lines generated by the domain walls.
This asymmetry is captured by a line conductivity, a physical observable of the
system. We prove that the line conductivity is quantized and given by the index
of a second Fredholm operator of Toeplitz type. We also prove a topological
charge conservation stating that the two aforementioned indices agree. This
result generalizes to higher dimensions and higher-order topological insulators
the bulk-edge correspondence of two-dimensional materials.
  We apply this procedure to evaluate the topological charge of several
classical examples of (standard and higher-order) topological insulators and
superconductors in one, two, and three spatial dimensions.
