Uncertainty Principles of Ingham and Paley-Wiener on Semisimple Lie Groups

TL;DR

This paper extends classical uncertainty principles by Ingham and Paley-Wiener to semisimple Lie groups, establishing new decay and support results, and applies these to unique continuation for Schrödinger equations on symmetric spaces.

Contribution

It develops analogues of classical Fourier uncertainty principles for semisimple Lie groups and demonstrates their application to Schrödinger equation unique continuation.

Findings

Established uncertainty principles on semisimple Lie groups.

Proved support and decay characterizations for Fourier transforms.

Showed unique continuation for Schrödinger equations on symmetric spaces.

Abstract

Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. Viewing these results as uncertainty principles for Fourier transforms, we prove certain analogues of these results on connected, noncompact, semisimple Lie groups with finite center. We also use these results to show unique continuation property of solutions to the initial value problem for time-dependent Schr\"odinger equations on Riemmanian symmetric spaces of noncompact type.