Unified approach for exactly solvable potentials in quantum mechanics using shift operators

Abstract

The local behavior of most solvable potentials reduces either to the harmonic oscillator or the Poschl-Teller potential. The general form for the commutation relations can be classified into two major categories, those belonging to the harmonic oscillator and those belonging to the Poschl-Teller potential. The study of solvable potentials in supersymmetric theory has deep physical connections with the inherent nonlinearity in the algebraic structure of the shift operators. The Morse and Poschl-Teller potentials do not provide analytical expression for the Franck-Codon factors.