Strichartz estimates for a Schr\"odinger equation on the half-line with a Neumann boundary condition

TL;DR

This paper establishes new Strichartz estimates for Schrödinger equations with Bessel operators on the half-line, and applies these results to prove well-posedness of certain nonlinear problems.

Contribution

It introduces novel Strichartz estimates for Bessel operators and a fractal restriction theorem, enabling analysis of nonlinear Schrödinger equations with boundary conditions.

Findings

Proved new Strichartz estimates for Bessel operators.

Established a fractal Tomas-Stein restriction theorem for the Hankel transform.

Demonstrated global well-posedness for critical nonlinear problems.

Abstract

In this paper we prove some new Strichartz estimates related to the Cauchy problem for the Bessel operator on the half-line and we establish a fractal version of the Tomas-Stein restriction theorem for the Hankel transform. Then we use the proved Strichartz estimates to show global in time well-posedness for a class of nonlinear $L^2_a$-critical problems, and local in time well-posedness in the sub-critical case.