  We describe a method for exactly diagonalizing clean $D$-dimensional lattice
systems of independent fermions subject to arbitrary boundary conditions in one
direction, as well as systems composed of two bulks meeting at a planar
interface. Our method builds on the generalized Bloch theorem [A. Alase et al.,
Phys. Rev. B 96, 195133 (2017)] and the fact that the bulk-boundary separation
of the Schrodinger equation is compatible with a partial Fourier transform
operation. Bulk equations may display unusual features because they are
relative eigenvalue problems for non-Hermitian, bulk-projected Hamiltonians.
Nonetheless, they admit a rich symmetry analysis that can simplify considerably
the structure of energy eigenstates, often allowing a solution in fully
analytical form. We illustrate our extension of the generalized Bloch theorem
to multicomponent systems by determining the exact Andreev bound states for a
simple SNS junction. We then analyze the Creutz ladder model, by way of a
conceptual bridge from one to higher dimensions. Upon introducing a new
Gaussian duality transformation that maps the Creutz ladder to a system of two
Majorana chains, we show how the model provides a first example of a
short-range chiral topological insulator hosting topological zero modes with a
power-law profile. Additional applications include the complete analytical
diagonalization of graphene ribbons with both zigzag-bearded and armchair
boundary conditions, and the analytical determination of the edge modes in a
chiral $p+ip$ two-dimensional topological superconductor. Lastly, we revisit
the phenomenon of Majorana flat bands and anomalous bulk-boundary
correspondence in a two-band gapless $s$-wave topological superconductor. We
analyze the equilibrium Josephson response of the system, showing how the
presence of Majorana flat bands implies a substantial enhancement in the
$4\pi$-periodic supercurrent.
