Trace formulas revisited and a new representation of KdV solutions with short-range initial data

TL;DR

This paper introduces a new approach to trace formulas for 1D Schrödinger operators with short-range potentials, demonstrating their invariance under KdV flow and applicability to dispersive smoothing effects.

Contribution

It presents a novel method for trace formulas that remain valid under KdV evolution, expanding the analytical tools for short-range potential analysis.

Findings

Trace formulas are preserved under KdV flow.

New representation of KdV solutions for short-range data.

Enhanced understanding of dispersive smoothing effects.

Abstract

We put forward a new approach to Deift-Trubowitz type trace formulas for the 1D Schrodinger operator with potentials that are summable with the first moment (short-range potentials). We prove that these formulas are preserved under the KdV flow whereas the class of short-range potentials is not. Finally, we show that our formulas are well-suited to study the dispersive smoothing effect.