Density of Schrödinger Weyl–Titchmarsh m functions on Herglotz functions

Abstract

We show that the Herglotz functions that arise as Weyl-Titchmarsh m functions of one-dimensional Schrödinger operators are dense in the space of all Herglotz functions with respect to uniform convergence on compact subsets of the upper half plane. This result is obtained as an application of de Branges theory of canonical systems.

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

“…In this section the m-functions for discrete Schrödinger operators are shown not to be dense on those for Jacobi operators. As mentioned in the introduction, this result is opposite to the density of Schrödinger m-functions on all Herglotz functions in [6] for the continuous setting.…”

“…In this paper, we would like to reveal the sparsity of discrete Schrödinger operators, which is contrary to the idea based on the density outcome of Schrödinger operators [6]. (In many applications we would expect to have the same conclusion on both continuous and discrete settings.)…”

“…In other words, there is only one H-integrable solution up to multiplicative constants, and therefore (2.6) is well-defined. See [6,15,16] for more details.…”

“…For this, a topology on canonical systems is required, such that it accords with the local uniform convergence on Herglotz functions. Besides de Branges' own work [2] for this (see also Lemma 2.1 in [14]), we adapt the following convergence on canonical systems in [6]: for all continuous functions f = (f 1 , f 2 ) t with compact support of [0, ∞).…”

“…Fortunately, in spite of this insufficiency, this set seems to be still ample in order to formulate an inverse spectral theory. One of the aspects showing this is the density of all their m-functions on the (compact) space of the Herglotz functions [6]. Schrödinger operators are enough to approximate any canonical systems.…”