The Darboux transformation and algebraic deformations of shape-invariant potentials

Abstract

We investigate the backward Darboux transformations (addition of the lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are simple elementary functions. A countable family, m = 0, 1,.2,..., of deformations exists for each family of shape-invariant potentials. We prove that the m_th deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules P_(m)^(m) Ϲ P_(-m+1)^(m) Ϲ (...) , where P_n^(m) is a codimension m subspace of <1, z,..., z_(n)>. In particular, we prove that the first (m = 1) algebraic deformation of the shape-invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules P_n^(1) = <1, z_(2),..., z_(n)>. By construction, these algebraically deformed Hamiltonians do not have an sl(2) hidden symmetry algebra structure.