Jacobi Matrices and de Branges Spaces

Scaling Limits of Jacobi Matrices and the Christoffel–Darboux Kernel

“…Remark 2.1. The process through which (2.8) is obtained is actually quite standard and shows that any Jacobi matrix may be associated with a discrete canonical system (see [28] for the analogue in the continuum and [29] for a different association). In fact, the matrix Q ℓ (x) and the difference equation it satisfies is a compact way of writing the variation of parameters method to express the eigenfunctions of J at x as varying linear combinations of the eigenfunctions at x 0 (see e.g.…”

“…Furthermore, the correspondence between such systems and Hermite-Biehler functions (see below for a definition), established by de Branges [8] is a central result in the theory of de Branges spaces. Thus there is a huge literature on canonical systems, spanning spectral theory, harmonic analysis and number theory ( [1,8,9,14,19,28,29] are only a few relevant references). The next several paragraphs contain only a quick a review of some results that are directly relevant here (for details, see e.g., [28]).…”

“…A canonical system is a family (indexed by z ∈ C) of differential equations of the form J u ′ (t) = zH(t)u(t) (6.1) on some interval I = [0, L] ⊆ R, where J = 0 −1 1 0 and H(t) is a 2 × 2 nonnegative definite matrix valued function such that the entries of H are integrable functions on I. By a change of variable (see, e.g., [28,Section 6] or [29]) we may also assume that H(t) ≡ 0 on nonempty open sets. In the case that H(t) is invertible almost everywhere, we may rewrite (6.1) as…”

“…While it is not an essential feature in the general approach, additional insight comes from the fact that these operators can be cast in the form of canonical systems [8,28,29]. It is well known that Jacobi matrices may be represented as particular cases of canonical systems and this representation has been shown to be useful in the context of inverse spectral theory ( [28] treats the analogous continuum case).…”

“…Moreover, scaling limits of CD kernels have been identified under certain conditions as reproducing kernels of de Branges spaces [25]. Such spaces are associated with canonical systems [8,28,29] through the fact that the underlying reproducing kernel can be obtained from the solutions to the corresponding eigenvalue equation. In the random setting, this approach has recently led to a random operator whose eigenvalue distribution corresponds to the asymptotic zero distribution on the critical line of the zeta function [37].…”