Calculation of the eigenvalues of Schrödinger equations by an extension of Hill's method

AbstractThe eigenfunctions of the one dimensional Schrödinger equation Ψ″ + [E − V(x)]Ψ=0, where V(x) is a polynomial, are represented by expansions of the form ∑k=0∞ckϕk(ω, x). The functions ϕk (ω, x) are chosen in such a way that recurrence relations hold for the coefficients ck: examples treated are Dk(ωx) (Weber-Hermite functions), exp (−ωx2)xk, exp (−cxq)Dk(ωx). From these recurrence relations, one considers an infinite bandmatrix whose finite square sections permit to solve approximately the original eigenproblem. It is then shown how a good choice of the parameter ω may reduce dramatically the complexity of the computations, by a theoretical study of the relation holding between the error on an eigenvalue, the order of the matrix, and the value of ω. The paper contains tables with 10 significant figures of the 30 first eigenvalues corresponding to V(x) = x2m, m = 2(1)7, and the 6 first eigenvalues corresponding to V(x) = x2 + λx10 and x2 + λx12, λ = .01(.01).1(.1)1(1)10(10)100