The Darboux transformation and algebraic deformations of shape-invariant potentials

الأصل · EN

We investigate the backward Darboux transformations (addition of a 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⁽ᵐ⁾ₘ₊₁⊂..., where P⁽ᵐ⁾ₙ is a codimension m subspace of <1,z,...,zⁿ>. 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⁽¹⁾ₙ = < 1,z²,...,zⁿ>. By construction, these algebraically deformed Hamiltonians do not have an sl(2) hidden symmetry algebra structure.