Computation of the spectra of linear differential operators. (Q1432655)

Evaluation of spectra of differential operators is involved in many applied problems. These problems may be linear and have the form \(Bv(x)=\mu v(x)\), where \(B\) is a differential operator. Quite often the differential operator \(B\) is approximated by a difference operator \(A\) and the algebraic problem \(A(u,\lambda)=0\) is solved instead of the original problem. In this paper, problems with linear operators \(B\) and \(A\) are considered. For the evaluation of spectra of linear differential operators of an arbitrary order, a grid method is suggested, which determines the spectrum of a matrix by inverse interations with shift. For second-order selfadjoint operators, such as Sturm-Liouville problems, in finite domains, the method of an augmented phase vector is more efficient. These techniques are illustrated with the help of some examples