  It has been shown that a positive semi-definite Hamiltonian H, that has a
tridiagonal matrix representation in a given basis, can be represented in the
form H = A{\dag}A, where A is a forward shift operator playing the role of an
annihilation operator. Such representation endows H with rich supersymmetric
properties yielding results analogous to those obtained by studying the
Hamiltonian as a differential operator. Here, we study the coherent states
which we define as being the eigenstates of the operator A. We explicitly find
the expansion coefficients of these states in the given basis. We further
identify a complete set of special coherent states which themselves can be used
as basis. In terms of these special coherent states, we show that a general
coherent state has the expansion form of a Lagrange interpolation scheme. As
application of the developed formalism, we work out examples of systems having
pure discrete, pure continuous, or mixed energy spectrum.
