We continue the study of the A-amplitude associated to a half-line Schrödinger operator, − d dx2 + q in L2((0, b)), b ≤ ∞. A is related to the WeylTitchmarshm-function via m(−κ2) = −κ− ∫ a 0 A(α)e −2ακ dα+O(e−(2a−ε)κ) for all ε > 0. We discuss five issues here. First, we extend the theory to general q in L1((0, a)) for all a, including q’s which are limit circle at infinity. Second, we prove th...

Small perturbations of the Jacobi matrix with weights √ n and zero diagonal are considered. A formula relating the asymptotics of polynomials of the first kind to the spectral density is obtained, which is analogue of the classical Weyl-Titchmarsh formula for the Schrödinger operator on the half-line with summable potential. Additionally a base of generalized eigenvectors for ”free” Hermite ope...

This paper touches upon several traditional topics of 1D linear complex analysis including distribution of zeros of entire functions, completeness problem for complex exponentials and for other families of special functions, some problems of spectral theory of selfadjoint differential operators. Their common feature is the close relation to the theory of complex Fourier transform of compactly s...

We discuss direct and inverse spectral theory of self-adjoint Sturm– Liouville relations with separated boundary conditions in the left-definite setting. In particular, we develop singular Weyl–Titchmarsh theory for these relations. Consequently, we apply de Branges’ subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to...

We provide a comparative treatment of some aspects of spectral theory for self-adjoint and non-self-adjoint (but J-self-adjoint) Dirac-type operators connected with the defocusing and focusing nonlinear Schrödinger equation, of relevance to nonlinear optics. In addition to a study of Dirac and Hamiltonian systems, we also introduce the concept of Weyl–Titchmarsh half-line m-coefficients (and 2 ...

Here γM (x) ∼ Q − 1 4 (x), QM (x) ∼ Q 1 2 (x) and ǫM (x) → 0 as x → ∞, while M(≥ 2) is an integer and γM and QM can be defined in terms of Q and its derivatives up to order M − 1. The general form of (1. 1) had been obtained previously by Cassell [5] [6] [7] and Eastham [10] [11, section 2.4]. In particular, the proof of (1. 1) in [10] and [11] depended on the formulation of (1. 2) as a first-o...

Based on continuity properties of the de Branges correspondence, we develop a new approach to study the high-energy behavior of Weyl–Titchmarsh and spectral functions of 2×2 first order canonical systems. Our results improve several classical results and solve open problems posed by previous authors. Furthermore, they are applied to radial Dirac and radial Schrödinger operators as well as to Kr...

We consider singular block operator problems of the type arising in the study of stability of the Ekman boundary layer. The essential spectrum is located, and an analysis of the L2 solutions of a related first order system of differential equations allows the development of a Titchmarsh–Weyl coefficientM (λ). This, in turn, permits a rigorous analysis of the convergence of approximations to the...