Spherical Schr\"odinger Hamiltonians: Spectral Analysis and Time Decay

TL;DR

This survey reviews recent spectral analysis results of spherical Schrödinger Hamiltonians, focusing on how initial data localization affects the decay of solutions in space $L^p$ norms, based on eigenvalues and eigenfunctions on the sphere.

Contribution

It provides a comprehensive description of the spectral properties and decay phenomena of spherical Schrödinger Hamiltonians, connecting eigenstructure to dispersive behavior.

Findings

Decay rates depend on frequency localization of initial data

Eigenvalues and eigenfunctions on the sphere determine decay behavior

Comparison with uncertainty inequalities offers insight into spectral properties

Abstract

In this survey, we review recent results concerning the canonical dispersive flow $e^{itH}$ led by a Schr\"odinger Hamiltonian $H$. We study, in particular, how the time decay of space $L^p$-norms depends on the frequency localization of the initial datum with respect to the some suitable spherical expansion. A quite complete description of the phenomenon is given in terms of the eigenvalues and eigenfunctions of the restriction of $H$ to the unit sphere, and a comparison with some uncertainty inequality is presented.