Pseudospectral methods for solving an equation of hypergeometric type with a perturbation

Almost all, regular or singular, Sturm–Liouville eigenvalue problems in the Schrödinger form − Ψ ″ ( x ) + V ( x ) Ψ ( x ) = E Ψ ( x ) , x ∈ ( a ̄ , b ̄ ) ⊆ R , Ψ ( x ) ∈ L 2 ( a ̄ , b ̄ ) for a wide class of potentials V ( x ) may be transformed into the form σ ( ξ ) y ″ + τ ( ξ ) y ′ + Q ( ξ ) y = − λ y , ξ ∈ ( a , b ) ⊆ R by means of intelligent transformations on both dependent and independent variables, where σ ( ξ ) and τ ( ξ ) are polynomials of degrees at most 2 and 1, respectively, and λ is a parameter. The last form is closely related to the equation of the hypergeometric type (EHT), in which Q ( ξ ) is identically zero. It will be called here the equation of hypergeometric type with a perturbation (EHTP). The function Q ( ξ ) may, therefore, be regarded as a perturbation. It is well known that the EHT has polynomial solutions of degree n for specific values of the parameter λ , i.e. λ : = λ n ( 0 ) = − n [ τ ′ + 1 2 ( n − 1 ) σ ″ ] , which form a basis for the Hilbert space L 2 ( a , b ) of square integrable functions. Pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of EHTP, and hence the energies E of the original Schrödinger equation. Specimen computations are performed to support the convergence numerically.