Frequency-dependent time decay of Schr\"odinger flows

TL;DR

This paper investigates how negative eigenvalues in the angular spectrum of electromagnetic Schr"odinger operators affect the time decay of solutions, revealing a nuanced interplay between spectral properties and dispersive behavior.

Contribution

It demonstrates that negative eigenvalues hinder classical time decay in Schr"odinger flows despite preserving dispersive estimates, and analyzes decay improvements for higher positive modes.

Findings

Negative eigenvalues cause lack of classical time decay.

Dispersive estimates still hold despite spectral issues.

Decay improves for higher positive eigenmodes.

Abstract

We show that the presence of negative eigenvalues in the spectrum of the angular component of an electromagnetic Schr\"odinger hamiltonian $H$ generically produces a lack of the classical time-decay for the associated Schr\"odinger flow $e^{-itH}$. This is in contrast with the fact that dispersive estimates (Strichartz) still hold, in general, also in this case. We also observe an improvement of the decay for higher positive modes, showing that the time decay of the solution is due to the first nonzero term in the expansion of the initial datum as a series of eigenfunctions of a quantum harmonic oscillator with a singular potential. A completely analogous phenomenon is shown for the heat semigroup, as expected.