  We study the essential self-adjointness of semi-bounded Schr\"{o}dinger
operators on birth-death chains. First, we offer a general characterization
which originates from studying a second order linear recurrence with
variational coefficients which comes from the Schr\"{o}dinger operator. As this
characterization is algebraically complicated, we present an additional theorem
discussing the failure of essential self-adjointness. Finally, we study two
specific cases of solutions to equations involving the Schr\"{o}dinger operator
over birth-death chains and derive explicit formulas in these cases.
