Dispersive estimates for Schr\"odinger operators with negative Coulomb-like potentials in one dimension

TL;DR

This paper establishes dispersive and Strichartz estimates for one-dimensional Schrödinger operators with Coulomb-like potentials, overcoming challenges posed by slow decay using spectral analysis and stationary phase techniques.

Contribution

It introduces a novel approach to derive dispersive estimates for slowly decaying potentials without relying on perturbation methods.

Findings

Proved dispersive estimates for Coulomb-like potentials in 1D.

Established orthonormal Strichartz estimates for the model.

Developed a spectral density expression using WKB approximation.

Abstract

In this paper, we consider the dispersive estimates for Schr\"odinger operators with Coulomb-like decaying potentials, such as $V(x)=-c|x|^{-\mu}$ for $|x|\gg 1$ with $0<\mu<2$, in one dimension. As an application, we establish both the standard and orthonormal Strichartz estimates for this model. One of the difficulties here is that perturbation arguments, which are typically applicable to rapidly decaying potentials, are not available. To overcome this, we derive a WKB expression for the spectral density and use a variant of the degenerate stationary phase formula to exploit its oscillatory behavior in the low-energy regime.