We consider a Sturm-Liouville boundary value problem in a bounded domain D of Rn. By this is meant that the differential equation is given by a second order elliptic operator of divergent form in D and the boundary conditions are of Robin type on ∂D. The first order term of the boundary operator is the oblique derivative whose coefficients bear discontinuities of the first kind. Applying the me...

It is shown that the operators associated with the perturbed wave equation in IR n and with the elliptic operators with an indeenite weight function and mildly varying coeecients on IR n are similar to a selfadjoint operator in a Hilbert space. These operators have the whole IR as the spectrum. It is shown that they are positive operators in corresponding Krein spaces, and the whole problem is ...

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov in...

We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov in...

We describe methods which have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. We find that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the opera...

The pseudospectra of non-selfadjoint linear ordinary differential operators with variable coefficients are considered. It is shown that when a certain winding number or twist condition is satisfied, closely related to Hörmander’s commutator condition for partial differential equations, 3-pseudoeigenfunctions of such operators for exponentially small values of 3 exist in the form of localized wa...

The combination of iterative methods with preconditionings based on incomplete LU factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations. In this paper, we show that there are some settings in which the incomplete LU preconditioners are not effective, and we demonstrate that their ...