Asymptotics of Titchmarsh--Weyl functions near the real axis andan application to the KdV hierarchy

Abstract

We establish high-energy asymptotic expansions of Titchmarsh--Weyl functions for one-dimensional Schr\"odinger and Dirac operators in regions whose boundaries approach the spectrum at a prescribed polynomial rate. For bounded potentials with bounded derivatives, the expansions remain uniform up to these boundaries, with an explicit loss in the remainder determined by the rate of approach. We treat both self-adjoint Dirac operators and non-self-adjoint Dirac operators with skew-adjoint potential matrices, keeping track of the distinct half-planes in which the two scalar Weyl coordinates are naturally defined. As an application of the Schr\"odinger expansion, we verify the high-energy hypothesis in Kotani's construction of KdV flows: for every odd integer $p\geq3$, each real-valued $q\in W^{2p-1,\infty}(\mathbb{R})$ generates a global classical solution of the member of the KdV hierarchy indexed by $(p+1)/2$.