Dispersion Estimates for Spherical Schr\"odinger Equations: The Effect of Boundary Conditions

TL;DR

This paper studies how boundary conditions at zero affect dispersive decay estimates for radial Schrödinger operators, revealing that boundary changes influence decay rates only for positive angular momentum, while negative or zero angular momentum maintains standard decay.

Contribution

It demonstrates the impact of boundary conditions on dispersive estimates for radial Schrödinger operators, highlighting differences based on angular momentum values.

Findings

Boundary condition changes affect decay estimates for positive angular momentum.

Standard decay rate of O(|t|^{-1/2}) holds for nonpositive angular momentum.

Dispersive decay estimates depend on boundary conditions only when angular momentum is positive.

Abstract

We investigate the dependence of the $L^1\to L^\infty$ dispersive estimates for one-dimensional radial Schr\"o\-din\-ger operators on boundary conditions at $0$. In contrast to the case of additive perturbations, we show that the change of a boundary condition at zero results in the change of the dispersive decay estimates if the angular momentum is positive, $l\in (0,1/2)$. However, for nonpositive angular momenta, $l\in (-1/2,0]$, the standard $O(|t|^{-1/2})$ decay remains true for all self-adjoint realizations.