A shooting method is developed to approximate the eigenvalues and eigenfunctions of a 4th order Sturm-Liouville problem. The main tool is a miss-distance function M(), which counts the number of eigenvalues less than. The method approximates the coeecients of the diierential equation by piecewise-constant functions, which enables an exact solution to be… (More)
This paper gives the analysis and numerics underlying a shooting method for approximating the eigenvalues of nonselfadjoint Sturm-Liouville problems. We consider even order problems with (equally divided) separated boundary conditions. The method can nd the eigenvalues in a rectangle and in a left half-plane. It combines the argument principle with the… (More)
Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M-function as an analytic function of a spectral parameter and the… (More)
The performance of Algorithm 700 is reviewed on a number of test problems, highlighting the strengths and weaknesses of the SLEIGN code.
1 The general problem class We consider on the interval (−π, π) the singular non-symmetric differential equation Lu := iε d dx f (x) du dx + i du dx = λu, (1.1) in which f is a 2π-periodic function having the following properties: f (x + π) = −f (x) , f (−x) = −f (x) ; (1.2) * We are grateful to Brian Davies for attracting our interest to the problem, and… (More)
We describe a new code (SLEUTH) for numerical solution of regular two-point fourth-order Sturm-Liouvlle eigenvalue problems. Eigenvalues are computed according to index: the user specifies an integer <italic>k</italic>***0, and the code computes an approximation to the <italic>k</italic>th eigenvalue. Eigenfunctions are also avialable through an auxiliary… (More)
We describe a new code (SLEUTH) for numerical solution of regular two-point fourth-order Sturm-Liouville eigenvalue problems. Eigenvalues are computed according to index: the user speciies an integer k 0 and the code computes an approximation to the kth eigen-value.Eigenfunctions are also available through an auxiliary routine, called after the eigenvalue… (More)
In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction A B of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces S and˜S such that the resolvent bordered by projections onto these… (More)