We systematically develop Weyl–Titchmarsh theory for singular differential operators on arbitrary intervals (a, b) ⊆ R associated with rather general differential expressions of the type τf = 1 r ( − ( p[f ′ + sf ] )′ + sp[f ′ + sf ] + qf ) , where the coefficients p, q, r, s are real-valued and Lebesgue measurable on (a, b), with p 6= 0, r > 0 a.e. on (a, b), and p−1, q, r, s ∈ Lloc((a, b); dx...

We study the trace class perturbations of half-line, discrete Laplacian and obtain a new bound for perturbation determinant corresponding non-self-adjoint Jacobi operator. Based on this bound, we Lieb–Thirring inequalities such operators. The spectral enclosure spectrum embedded eigenvalues are also discussed.

Implicit–explicit multistep methods for nonlinear parabolic equations were recently analyzed in [2, 3, 1]. In these papers the linear operator of the equation is assumed time-independent, self-adjoint and positive definite; then, the linear part is discretized implicitly and the remaining part explicitly. Here we slightly relax the hypotheses on the linear operator by allowing part of it to be ...

We consider operator-valued Herglotz functions and their applications to self-adjoint perturbations of self-adjoint operators and self-adjoint extensions of densely defined closed symmetric operators. Our applications include model operators for both situations, linear fractional transformations for Herglotz operators, results on Friedrichs and Krein extensions, and realization theorems for cla...

We consider a singular Sturm-Liouville differential expression with an indefinite weight function and we show that the corresponding self-adjoint differential operator in a Krein space locally has the same spectral properties as a definitizable operator.

The one-dimensional Dirac operator with a singular interaction term which is formally given by A?|?0???0|, where A an arbitrary 2×2 matrix and ?0 stands for the distribution, introduced as closed not necessarily self-adjoint operator. We study its spectral properties, find non-relativistic limit also address question of regular approximations. In particular, we show that, contrary to case local...

It is shown that the dynamical evolution of perturbations on a static spacetime is governed by a standard pulsation equation for the extrinsic curvature tensor. The centerpiece of the pulsation equation is a wave operator whose spatial part is manifestly self-adjoint. In contrast to metric formulations, the curvature-based approach to perturbation theory generalizes in a natural way to self-gra...

Among all linear operators on Hilbert spaces, the compact ones (defined below) are the simplest, and most imitate the more familiar linear algebra of finite-dimensional operator theory. In addition, these are of considerable practical value and importance. We prove a spectral theorem for self-adjoint operators with minimal fuss. Thus, we do not invoke broader discussions of properties of spectr...

Let à be a self-adjoint extension in K̃ of a fixed symmetric operator A in K ⊆ K̃. An analytic characterization of the eigenvalues of à is given in terms of the Q-function and the parameter function in the Krein–Naimark formula. Here K and K̃ are Krein spaces and it is assumed that à locally has the same spectral properties as a self-adjoint operator in a Pontryagin space. The general results are ...