  We develop a theory of edge states based on the Hermiticity of Hamiltonian
operators for tight-binding models defined on lattices with boundaries. We
describe Hamiltonians using shift operators which serve as differential
operators in continuum theories. It turns out that such Hamiltonian operators
are not necessarily Hermitian on lattices with boundaries, which is due to the
boundary terms associated with the summation by parts. The Hermiticity of
Hamiltonian operators leads to natural boundary conditions, and for models with
nearest-neighbor (NN) hoppings only, there are reference states that satisfy
the Hermiticity and boundary conditions simultaneously. Based on such reference
states, we develop a Bloch-type theory for edge states of NN models on a
half-plane. This enables us to extract Hamiltonians describing edge-states at
one end, which are separated from the bulk contributions. It follows that we
can describe edge states at the left and right ends separately by distinct
Hamiltonians for systems of cylindrical geometry. We show various examples of
such edge state Hamiltonians (ESHs), including Hofstadter model, graphene
model, and higher-order topological insulators (HOTIs).
