Essential Self-Adjointness of Semi-Bounded Schrodinger Operators on Birth-Death Chains

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.