Coherent states associated with tridiagonal Hamiltonians

It has been shown that a positive semi-definite Hamiltonian H, that has a tridiagonal matrix representation in a given basis $\left\{|{\varphi }_n〉\right\}$, can be represented in the form H = A^†A, where A is a forward shift operator satisfying $A|{\varphi }_n〉=c_n|{\varphi }_n〉+d_n|{\varphi }_n-1〉$ playing the role of an annihilation operator. Here, the coherent states |z) are defined as eigenstates of A. We also exhibit a complete set of coherent states {|c_n)}, labeled by the discrete points c_n, which we may also use as a basis. We find a solution of the coherent state |z), indexed by the continuous parameter z as a series expansion in terms of the {|c_n)}. We further show how to compute the time development of the coherent state |z) and we illustrate this with examples. As a major result, we find the explicit closed form solution |z) for the Morse oscillator.