Continuous Analogue of Sturm Sequences in the Context of Sturm-Liouville Equations. Revision

摘要

A shooting method is presented for finding the n sub th eigenvalue and eigenfunction of a Sturm-Liouville equation, in which the eigenvalue occurs nonlinearly. The method is verified in two ways: by applying the Sturm comparison and oscillation theorems to the continuous problem; and by applying Sturm sequences to a discretization. The method works for general (separated) boundary conditions, and provides an a-posteriori error estimate for the approximate eigenvalue. Analogues of the Sturm comparison, oscillation and separation theorems are proved for the discrete problem. A related method, which involves critical lengths in the invariant imbedding method, is shown to be incorrect for general boundary conditions.