A counter example on nontangential convergence for oscillatory integrals

TL;DR

This paper demonstrates that for certain operators, the convergence properties of solutions to the Schrödinger equation do not improve beyond known limitations, extending previous counterexamples to more general operators.

Contribution

It generalizes existing counterexamples of non-tangential convergence failure from the standard Schrödinger operator to a broader class of operators   with specific conditions.

Findings

Counterexample applies to a wider class of operators   with specific conditions.

Non-tangential convergence cannot be extended for these operators.

Results highlight limitations of convergence regions for solutions of generalized Schrödinger equations.

Abstract

Consider the solution of the time-dependent Schr{\"o}dinger equation with initial data $f$. It is shown in \cite{artikel} that there exists $f$ in the Sobolev space $H^s(\RR), s=n/2$ such that tangential convergence can not be widened to convergence regions. In this paper we show that the corresponding result holds when $-\Delta_x$ is replaced by an operator $\phi(D)$, with special conditions on $\phi$.