Generalized free time-dependent Schr\"odinger equation with initial data in Fourier Lebesgue spaces

TL;DR

This paper extends the analysis of the Schrödinger equation's convergence properties to initial data in Fourier Lebesgue spaces, generalizing previous results from Sobolev spaces and operators.

Contribution

It demonstrates that convergence limitations for the Schrödinger equation also hold when initial data are in Fourier Lebesgue spaces, broadening the scope of prior results.

Findings

Convergence cannot be widened for initial data in Fourier Lebesgue spaces.

Results generalize previous Sobolev space findings.

Applicable to generalized Schrödinger equations with operator (D).

Abstract

Consider the solution of the free time-dependent Schr\"odinger equation with initial data f. It is shown by Sj\"ogren and Sj\"olin (1989) that there exists f in the Sobolev space H^s(R^d), s=d/2 such that tangential convergence can not be widened to convergence regions. In 2010 we obtained the corresponding results for a generalized version of the Schr\"odinger equation, where -\Delta_x is replaced by an operator \phi(D), with special conditions on \phi. In this paper we show that similar results may be obtained for initial data in Fourier Lebesgue spaces.