Relative oscillation theory for Jacobi matrices extended

Abstract

Abstract. We present a comprehensive treatment of relative oscillation theory for finite Jacobi matrices. We show that the difference of the number of eigenvalues of two Jacobi matrices in an interval equals the number of weighted sign-changes of the Wronskian of suitable solutions of the two underlying difference equations. Until now only the case of perturbations of the main diagonal was known. We extend the known results to arbitrary perturbations, allow any (half-) open and closed spectral intervals, simplify the proof, and establish the comparison theorem.Mathematics subject classification (2010): Primary 39A21, 47B36; Secondary 34C10, 34L05.

Renormalized oscillation theory for Hamiltonian systems

“…Reference [14] also contains results with ψ + (λ 1 , • ) replaced by ψ − (λ 1 , • ), and other extensions, particularly, to self-adjoint, separated boundary conditions, but we omit further details here. In addition, extensions of Theorem 1.2, as well as the treatment of Dirac-type operators and that of the finite difference case of Jacobi operators appeared in [1], [33], [34], [35], [44], [46]- [48], [49,Ch. 4].…”

Spectral theory as influenced by Fritz Gesztesy

“…In this respect we should also mention the beautiful result by Fritz and Ünal [78] which gives the most general version of Kneser's theorem. Extensions to other operators can be found in [4,29,126,127,130].…”

Discrete oscillation theorems and weighted focal points for Hamiltonian difference systems with nonlinear dependence on a spectral parameter

“…For this purpose we introduce a new notion of weighted focal points for a conjoined basis of (1.1). This notion is closely related to the notions of weighted nodes (for n = 1) and relative oscillation numbers (for n > 1) introduced for the Wronskians of solutions of the scalar and matrix difference Sturm-Liouville equations (see [9,11,12]). As it is shown in [10], the classical function n 1 (λ) of the number of focal points in (0, N + 1] can lose the monotonic character with respect to λ if (1.10) is not satisfied.…”