A remark on the Schr\"odinger equation on Zoll manifolds

TL;DR

This paper investigates the singularity structure of solutions to the Schrödinger equation on Zoll manifolds, extending known results from spheres and symmetric spaces by analyzing the functional calculus of the associated self-adjoint operator.

Contribution

It extends the analysis of Schrödinger equation singularities from spheres and symmetric spaces to Zoll manifolds, providing new insights into their spectral properties.

Findings

Support of solutions concentrated at specific times

Singularity behavior characterized by eigenvalue analysis

Extension of known results to Zoll manifolds

Abstract

On the one dimensional sphere, the support of the fundamental solution to the Schr$\rm \ddot o$dinger equation consists of finite points at the time $t\in 2\pi\Q$. The paper \cite{Ka} generalized this fact to compact symmetric spaces. In this paper, we consider similar results on Zoll manifolds. We study the singularity for a solution to the equation using a functional calculus of the self-adjoint operator with integer eigenvalues.