Sobolev mapping properties of the scattering transform for the Schr\"odinger equation
Rostyslav O. Hryniv, Yaroslav V. Mykytyuk, Peter A. Perry
TL;DR
This paper investigates the mathematical properties of the scattering transform for the Schrödinger equation with singular potentials, establishing invertibility and Lipschitz continuity in specific function spaces using Riccati representation.
Contribution
It introduces a novel analysis of the scattering transform's properties for singular potentials without bound states, connecting Schrödinger and ZS-AKNS scattering theories.
Findings
Invertibility of the scattering transform in weighted and Sobolev spaces
Lipschitz continuity of the scattering transform
Connection established between Schrödinger and ZS-AKNS scattering theories
Abstract
We consider the scattering transform for the Schr\"odinger equation with a singular potential and no bound states. Using the Riccati representation for real-valued potentials on the line, we obtain invertibility and Lipschitz continuity of the scattering transform between weighted and Sobolev spaces. Our approach exploits the connection between scattering theory for the Schr\"odinger equation and scattering theory for the ZS-AKNS system
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Taxonomy
TopicsAdvanced Mathematical Physics Problems · Spectral Theory in Mathematical Physics · Numerical methods in inverse problems