An Alternate Proof of the De Branges Theorem on Canonical Systems

Abstract

The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by canonical systems, is constant on ℂ. This provides an alternative proof of the De Branges theorem that the canonical systems with tr H1 imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a canonical system.

The m-functions of discrete Schrödinger operators are sparse compared to those for Jacobi operators

“…Such a solution satisfying (2.7) is called H-integrable. Note that a half-line trace-normed canonical system is always in a limit-point case at ∞, which was obtained from the original argument by [2] or an alternative proof in [1]. In other words, there is only one H-integrable solution up to multiplicative constants, and therefore (2.6) is well-defined.…”

“…Note that the trace-normed condition, Tr H(t) = 1, is preserved in the limiting process, which indicates that H(t)dt is absolutely continuous with respect to the Lebesgue measure. Since there is no point spectrum for these measures, any vectors containing characteristic functions, such as (χ [1,2) (t), 0) t , can be test functions in Lemma 4.1. In other words, the weak- * convergence may be operated with any characteristic functions.…”

“…In other words, its second and third steps look like ones on the following figure: We now see that discrete Schrödinger canonical systems cannot approach the Jacobi canonical system. Recall first that, due to (3.13), any ϕ DS for discrete Schrödinger canonical systems should be constant at least on [1,2).…”

“…Let us focus on the interval [1,2). Since any constant function cannot be close to two different steps at the same time in the L 1 -sense, Hamiltonians of discrete Schrödinger canonical systems cannot converge in weak- * to the one for the given Jacobi canonical system.…”

Density of Schrödinger Weyl–Titchmarsh m functions on Herglotz functions

“…Our goal in this paper is to present a transparent way to avoid these difficulties and, therefore, answer the problem positively. The crucial machinery for the new path is de Branges theory of canonical systems [1,6,7,8,9,10,12,17,19,28,32,37,38], which enables us to rephrase the problem as one about approximations of canonical systems.…”