Dynamic representation of the Weyl solution for the Schrödinger operator on the semi-axis

TL;DR

This work develops a new representation for the Weyl solution of the Schrödinger operator on the half-line by linking spectral data to the dynamical problem for the wave equation with the same potential. The authors express the wave-field with boundary control via a kernel solving a Goursat problem, recast it as a integral equation, and propagate solvability and bounds to obtain a Weyl-solution representation through Fourier analysis. The results cover $q\in L^1(\mathbb{R}_+)$ with a spectral-region $\operatorname{Im}k>\|q\|_{L^1}/4$ and extend to $q\in l^\infty(L^1(\mathbb{R}_+))$ with explicit Neumann-series convergence conditions, involving the transformed potential $\tilde q(\gamma)=q(\gamma/2)$. This provides a dynamical, constructive perspective on Weyl functions, clarifying the link to the A-amplitude/response function framework and broadening applicability beyond prior Laplace-transform approaches.

Abstract

We derive a representation formula for the Weyl solution to the Schrödinger operator on the semi-axis for certain classes of potentials. Our approach is based on relations with the initial-boundary value problem for the wave equation with the same potential on the half-line.

Theorems & Definitions (5)

• Proposition 1
• Proposition 2
• Proposition 3
• Lemma 1
• Proposition 4