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

摘要

Almost all, regular or singular, Sturm–Liouville eigenvalue problems in the Schrödinger form −Ψ″(x)+V(x)Ψ(x)=EΨ(x),x∈(ā,b̄)⊆R,Ψ(x)∈L2(ā,b̄) for a wide class of potentials V(x) may be transformed into the form σ(ξ)y″+τ(ξ)y′+Q(ξ)y=−λy,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:=λn(0)=−n[τ′+12(n−1)σ″], which form a basis for the Hilbert space L2(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.