A theory of shift operators with applications to nonharmonic systems

We present a theory of shift operators (i.e., operators which shift given solutions into other solutions), including their relationship with deformed algebras and describe a general constructive method which enables us to calculate such operators for a wide class of problems. These include the classical linear differential equations of the hypergeometric and confluent hypergeometric functions, a number of soluble nonrelativistic Schrödinger equations (including one with a non-Hermitian Hamiltonian), and a simple master equation. In general, the resulting shift-up and shift-down operators are level dependent but allow for the sequential generation of all required solutions.