Riccati equation, factorization method and shape invariance (Q2730403)

Two linear operators \(H\) and \(\widetilde H\) are said to be \(A\)-related if there exists an operator \(A\) such that \(AH={\widetilde H}A\). If we assume that the operator \(A\) relating \(H\) and \(\widetilde H\) is a first order differential operator of the form \(A={d\over{dx}}+W(x),\) then the relation \(AH=\widetilde H A\), with \(H=-{{d^2}\over{dx^2}}+V(x)\), \(\widetilde H=-{{d^2}\over{dx^2}}+\widetilde V(x),\) leads to the pair of Riccati equations \(V-d=W^2-W'\), \(\widetilde V-d=W^2+W'\), \(d= \text{const}\). Moreover, both operators \(H\) and \(\widetilde H\) can be factorized as \(H=A^\dagger A+d\), \(\widetilde H=A A^\dagger+d\). NEWLINENEWLINENEWLINEThe factorization method was introduced by \textit{E. Schrödinger} [Proc. R. Irish Acad. A 46, 9-16 (1940; Zbl 0023.08602); 46, 183-206 (1941; Zbl 0063.06818); 47, 53-54 (1941; Zbl 0063.06821)] and developed by \textit{L. Infeld} and \textit{T. E. Hull} [Rev. Mod. Phys. 23, 21-68 (1951; Zbl 0043.38602)]. This method is very efficient in the search of exactly solvable potentials. NEWLINENEWLINENEWLINEAssume that \(V, \widetilde V\) depend on a certain set of parameters \(a\) and \(\widetilde V(x,a) = V(x,f(a)) + R(f(a))\), where \(f\) is an invertible and differentiable transformation over the set of parameters. This concept introduced by \textit{L. E. Gendenshteĭn} [JETP Lett. 38, 356 (1983)[ is usually called ``shape invariance''. NEWLINENEWLINENEWLINEThe authors review the basic concepts of factorizable problems in one-dimensional Quantum Mechanics, as well as the theory of shape invariant potentials. The relation of this last theory with a generalization of the classical factorization method presented by Infeld and Hull is analyzed in detail. By the use of some properties of the Riccati equation the solutions of Infeld and Hull are generalized.