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

TL;DR

This paper investigates the conditions under which semi-bounded Schr"odinger operators on birth-death chains are essentially self-adjoint, providing a general characterization, conditions for failure, and explicit solutions in specific cases.

Contribution

It introduces a new algebraic characterization for essential self-adjointness of these operators and analyzes particular cases with explicit formulas.

Findings

General algebraic characterization of essential self-adjointness

Conditions under which self-adjointness fails

Explicit solutions for specific birth-death chain cases

Abstract

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.