The shooting method for solving eigenvalue problems
The generalized eigenvalue problem \({d^2R\over dv^2}=Q(v,\lambda)R\), \(v\in (-\infty,\infty)\), \(\lambda\in [a,b]\) under a boundedness condition is considered. The main results concern a characterization of the eigenvalues. Let \(Z(\lambda)\) be the number of zeros of a solution bounded on \((0,\infty)\) for \(\lambda\in [a,b]\). The author proves that under certain conditions on \(Q\), \(Z(\lambda)\) is well defined, nondecreasing and \(\lambda\) is an eigenvalue iff it is a discontinuity of \(Z(\lambda)\). The obtained results give a theoretical basis for the shooting method for solving the eigenvalue problem. The author presents applications of the shooting method to the radial equation associated with the \(n\)-dimensional Schrödinger equation with spherically symmetric potential function.
scientific article; zbMATH DE number 944363